Integrand size = 25, antiderivative size = 565 \[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {a^2 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d \left (19 b c^2-15 a d^2\right ) x\right )}{6 b^3 \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}+\frac {14 c \sqrt {c+d x} \sqrt {a-b x^2}}{15 b^3 d^2}-\frac {2 (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b^3 d^2}-\frac {\sqrt {a} \left (32 b^3 c^6+92 a b^2 c^4 d^2-375 a^2 b c^2 d^4+231 a^3 d^6\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{30 b^{7/2} d^3 \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {a} c \left (32 b^2 c^4+116 a b c^2 d^2-153 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{30 b^{7/2} d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
1/3*a^3*(-d*x+c)*(d*x+c)^(1/2)/b^3/(-a*d^2+b*c^2)/(-b*x^2+a)^(3/2)-1/6*a^2 *(d*x+c)^(1/2)*(2*c*(-7*a*d^2+9*b*c^2)-d*(-15*a*d^2+19*b*c^2)*x)/b^3/(-a*d ^2+b*c^2)^2/(-b*x^2+a)^(1/2)+14/15*c*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^3/d^ 2-2/5*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^3/d^2-1/30*a^(1/2)*(231*a^3*d^6-375 *a^2*b*c^2*d^4+92*a*b^2*c^4*d^2+32*b^3*c^6)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2 )*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^ (1/2)*c+a^(1/2)*d))^(1/2))/b^(7/2)/d^3/(-a*d^2+b*c^2)^2/(b^(1/2)*(d*x+c)/( b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+1/30*a^(1/2)*c*(-153*a^2*d^4+ 116*a*b*c^2*d^2+32*b^2*c^4)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)* (1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2 )*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(7/2)/d^3/(-a*d^2+b*c^2)/(d*x +c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.18 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.35 \[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (\frac {16 c}{d^2}-\frac {12 x}{d}+\frac {10 a^3 (c-d x)}{\left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}+\frac {5 a^2 \left (b c^2 (18 c-19 d x)+a d^2 (-14 c+15 d x)\right )}{\left (b c^2-a d^2\right )^2 \left (-a+b x^2\right )}\right )}{b^3}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (32 b^3 c^6+92 a b^2 c^4 d^2-375 a^2 b c^2 d^4+231 a^3 d^6\right ) \left (a-b x^2\right )+i \sqrt {b} \left (32 b^{7/2} c^7-32 \sqrt {a} b^3 c^6 d+92 a b^{5/2} c^5 d^2-92 a^{3/2} b^2 c^4 d^3-375 a^2 b^{3/2} c^3 d^4+375 a^{5/2} b c^2 d^5+231 a^3 \sqrt {b} c d^6-231 a^{7/2} d^7\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (32 b^3 c^6-8 \sqrt {a} b^{5/2} c^5 d+92 a b^2 c^4 d^2+106 a^{3/2} b^{3/2} c^3 d^3-375 a^2 b c^2 d^4-78 a^{5/2} \sqrt {b} c d^5+231 a^3 d^6\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b^4 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2 d-a d^3\right )^2 \left (-a+b x^2\right )}\right )}{30 \sqrt {c+d x}} \] Input:
Integrate[x^7/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*((16*c)/d^2 - (12*x)/d + (10*a^3*(c - d*x))/( (b*c^2 - a*d^2)*(a - b*x^2)^2) + (5*a^2*(b*c^2*(18*c - 19*d*x) + a*d^2*(-1 4*c + 15*d*x)))/((b*c^2 - a*d^2)^2*(-a + b*x^2))))/b^3 + (d^2*Sqrt[-c + (S qrt[a]*d)/Sqrt[b]]*(32*b^3*c^6 + 92*a*b^2*c^4*d^2 - 375*a^2*b*c^2*d^4 + 23 1*a^3*d^6)*(a - b*x^2) + I*Sqrt[b]*(32*b^(7/2)*c^7 - 32*Sqrt[a]*b^3*c^6*d + 92*a*b^(5/2)*c^5*d^2 - 92*a^(3/2)*b^2*c^4*d^3 - 375*a^2*b^(3/2)*c^3*d^4 + 375*a^(5/2)*b*c^2*d^5 + 231*a^3*Sqrt[b]*c*d^6 - 231*a^(7/2)*d^7)*Sqrt[(d *(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] /Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt [a]*Sqrt[b]*d*(32*b^3*c^6 - 8*Sqrt[a]*b^(5/2)*c^5*d + 92*a*b^2*c^4*d^2 + 1 06*a^(3/2)*b^(3/2)*c^3*d^3 - 375*a^2*b*c^2*d^4 - 78*a^(5/2)*Sqrt[b]*c*d^5 + 231*a^3*d^6)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]* d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(b^4*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2*d - a*d^3)^2 *(-a + b*x^2))))/(30*Sqrt[c + d*x])
Time = 1.87 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {602, 27, 2180, 27, 2185, 27, 2185, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^7}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 602 |
| \(\displaystyle \frac {\int \frac {-6 a \left (c^2-\frac {a d^2}{b}\right ) x^5-\frac {6 a^2 \left (b c^2-a d^2\right ) x^3}{b^2}-\frac {3 a^3 \left (2 b c^2-a d^2\right ) x}{b^3}+\frac {a^4 c d}{b^3}}{2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-6 a \left (c^2-\frac {a d^2}{b}\right ) x^5-\frac {6 a^2 \left (b c^2-a d^2\right ) x^3}{b^2}-\frac {3 a^3 \left (2 b c^2-a d^2\right ) x}{b^3}+\frac {a^4 c d}{b^3}}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2180 |
| \(\displaystyle \frac {\frac {\int -\frac {\frac {2 c d \left (9 b c^2-7 a d^2\right ) a^4}{b^3}-\frac {\left (24 b^2 c^4-67 a b d^2 c^2+39 a^2 d^4\right ) x a^3}{b^3}-\frac {12 \left (b c^2-a d^2\right )^2 x^3 a^2}{b^2}}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\frac {2 c d \left (9 b c^2-7 a d^2\right ) a^4}{b^3}-\frac {\left (24 b^2 c^4-67 a b d^2 c^2+39 a^2 d^4\right ) x a^3}{b^3}-\frac {12 \left (b c^2-a d^2\right )^2 x^3 a^2}{b^2}}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {2 \int \frac {\frac {2 c d^2 \left (18 b^2 c^4-81 a b d^2 c^2+53 a^2 d^4\right ) a^3}{b^2}-\frac {84 c d^2 \left (b c^2-a d^2\right )^2 x^2 a^2}{b}-d \left (24 b c^6-204 a d^2 c^4+\frac {431 a^2 d^4 c^2}{b}-\frac {231 a^3 d^6}{b^2}\right ) x a^2}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\int \frac {\frac {2 c d^2 \left (18 b^2 c^4-81 a b d^2 c^2+53 a^2 d^4\right ) a^3}{b^2}-\frac {84 c d^2 \left (b c^2-a d^2\right )^2 x^2 a^2}{b}-d \left (24 b c^6-204 a d^2 c^4+\frac {431 a^2 d^4 c^2}{b}-\frac {231 a^3 d^6}{b^2}\right ) x a^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}-\frac {2 \int -\frac {3 a^2 d^3 \left (2 a c d \left (4 b c^4-53 a d^2 c^2+\frac {39 a^2 d^4}{b}\right )+\left (32 b^2 c^6+92 a b d^2 c^4-375 a^2 d^4 c^2+\frac {231 a^3 d^6}{b}\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \int \frac {2 a c d \left (4 b c^4-53 a d^2 c^2+\frac {39 a^2 d^4}{b}\right )+\left (32 b^2 c^6+92 a b d^2 c^4-375 a^2 d^4 c^2+\frac {231 a^3 d^6}{b}\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \left (\frac {\left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}\right )}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}\right )}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \left (-\frac {c \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \left (-\frac {c \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{b d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {a^2 d \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{b^{3/2} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b}+\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}+\frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {a^3 (c-d x) \sqrt {c+d x}}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}+\frac {-\frac {a^3 \sqrt {c+d x} \left (2 c \left (9 b c^2-7 a d^2\right )-d x \left (19 b c^2-15 a d^2\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {24 a^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}-\frac {\frac {56 a^2 c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^2}+\frac {a^2 d \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (-153 a^2 d^4+116 a b c^2 d^2+32 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{b^{3/2} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {231 a^3 d^6}{b}-375 a^2 c^2 d^4+92 a b c^4 d^2+32 b^2 c^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
Input:
Int[x^7/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(a^3*(c - d*x)*Sqrt[c + d*x])/(3*b^3*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) + (-((a^3*Sqrt[c + d*x]*(2*c*(9*b*c^2 - 7*a*d^2) - d*(19*b*c^2 - 15*a*d^2)*x ))/(b^3*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - ((24*a^2*(b*c^2 - a*d^2)^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b^3*d^2) - ((56*a^2*c*d*(b*c^2 - a*d^2)^2 *Sqrt[c + d*x]*Sqrt[a - b*x^2])/b^2 + (a^2*d*((-2*Sqrt[a]*(32*b^2*c^6 + 92 *a*b*c^4*d^2 - 375*a^2*c^2*d^4 + (231*a^3*d^6)/b)*Sqrt[c + d*x]*Sqrt[1 - ( b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/( (Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(b*c^2 - a*d^2)*(32*b^2*c^4 + 116*a*b*c^2*d^2 - 153*a^2*d^4)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqr t[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a] ]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(b^(3/2)*d*Sqrt[c + d*x]*Sqr t[a - b*x^2])))/b)/(5*b*d^3))/(2*a*(b*c^2 - a*d^2)))/(6*a*(b*c^2 - a*d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomia lRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^(p + 1)*((a*(d*e - c*f) + (b*c*e + a*d*f)*x)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2 *a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToS um[2*a*(p + 1)*(b*c^2 + a*d^2)*Qx + e*(b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3 )) - a*c*d*f*n + d*(b*c*e + a*d*f)*(n + 2*p + 4)*x, x], x], x]] /; FreeQ[{a , b, c, d, n}, x] && IGtQ[m, 1] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* (p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p + 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e , m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && !(IGtQ[ m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 13.77 (sec) , antiderivative size = 962, normalized size of antiderivative = 1.70
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {\left (\frac {d \,a^{3} x}{3 \left (a \,d^{2}-b \,c^{2}\right ) b^{5}}-\frac {c \,a^{3}}{3 b^{5} \left (a \,d^{2}-b \,c^{2}\right )}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{\left (x^{2}-\frac {a}{b}\right )^{2}}-\frac {2 \left (-b d x -b c \right ) \left (-\frac {a^{2} d \left (15 a \,d^{2}-19 b \,c^{2}\right ) x}{12 b^{4} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {a^{2} c \left (7 a \,d^{2}-9 b \,c^{2}\right )}{6 b^{4} \left (a \,d^{2}-b \,c^{2}\right )^{2}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {2 x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b^{3} d}+\frac {8 c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{15 b^{3} d^{2}}+\frac {2 \left (-\frac {c d \,a^{2}}{6 b^{3} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {d \,a^{2} c \left (7 a \,d^{2}-9 b \,c^{2}\right )}{6 b^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {c \,a^{2} d \left (15 a \,d^{2}-19 b \,c^{2}\right )}{6 b^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {2 a c}{15 b^{3} d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {13 a}{5 b^{3}}+\frac {a^{2} d^{2} \left (15 a \,d^{2}-19 b \,c^{2}\right )}{12 b^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {8 c^{2}}{15 b^{2} d^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(962\) |
| risch | \(\text {Expression too large to display}\) | \(1953\) |
| default | \(\text {Expression too large to display}\) | \(4189\) |
Input:
int(x^7/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((1/3*d/(a*d^2-b *c^2)/b^5*a^3*x-1/3*c/b^5*a^3/(a*d^2-b*c^2))*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^ (1/2)/(x^2-a/b)^2-2*(-b*d*x-b*c)*(-1/12*a^2*d*(15*a*d^2-19*b*c^2)/b^4/(a*d ^2-b*c^2)^2*x+1/6*a^2*c*(7*a*d^2-9*b*c^2)/b^4/(a*d^2-b*c^2)^2)/((x^2-a/b)* (-b*d*x-b*c))^(1/2)-2/5/b^3/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+8/15/b^ 3/d^2*c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-1/6*c*d/b^3*a^2/(a*d^2-b*c^ 2)-1/6/b^3*d*a^2*c*(7*a*d^2-9*b*c^2)/(a*d^2-b*c^2)^2+1/6/b^3*c*a^2*d*(15*a *d^2-19*b*c^2)/(a*d^2-b*c^2)^2+2/15/b^3/d*a*c)*(c/d-1/b*(a*b)^(1/2))*((x+c /d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2 )))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c *x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c /d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(13/5*a/b^3+1/12*a^2* d^2*(15*a*d^2-19*b*c^2)/b^3/(a*d^2-b*c^2)^2+8/15/b^2/d^2*c^2)*(c/d-1/b*(a* b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d -1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2 )/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x +c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b) ^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^ (1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))))
Time = 0.16 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.50 \[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(x^7/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/90*((32*a^2*b^3*c^7 + 68*a^3*b^2*c^5*d^2 - 57*a^4*b*c^3*d^4 - 3*a^5*c*d^ 6 + (32*b^5*c^7 + 68*a*b^4*c^5*d^2 - 57*a^2*b^3*c^3*d^4 - 3*a^3*b^2*c*d^6) *x^4 - 2*(32*a*b^4*c^7 + 68*a^2*b^3*c^5*d^2 - 57*a^3*b^2*c^3*d^4 - 3*a^4*b *c*d^6)*x^2)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 3*(32*a^2*b^3*c^6 *d + 92*a^3*b^2*c^4*d^3 - 375*a^4*b*c^2*d^5 + 231*a^5*d^7 + (32*b^5*c^6*d + 92*a*b^4*c^4*d^3 - 375*a^2*b^3*c^2*d^5 + 231*a^3*b^2*d^7)*x^4 - 2*(32*a* b^4*c^6*d + 92*a^2*b^3*c^4*d^3 - 375*a^3*b^2*c^2*d^5 + 231*a^4*b*d^7)*x^2) *sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/ 27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(16*a^2*b^3*c^5*d^ 2 - 112*a^3*b^2*c^3*d^4 + 76*a^4*b*c*d^6 - 12*(b^5*c^4*d^3 - 2*a*b^4*c^2*d ^5 + a^2*b^3*d^7)*x^5 + 16*(b^5*c^5*d^2 - 2*a*b^4*c^3*d^4 + a^2*b^3*c*d^6) *x^4 + (24*a*b^4*c^4*d^3 - 143*a^2*b^3*c^2*d^5 + 99*a^3*b^2*d^7)*x^3 - 2*( 16*a*b^4*c^5*d^2 - 77*a^2*b^3*c^3*d^4 + 51*a^3*b^2*c*d^6)*x^2 - (12*a^2*b^ 3*c^4*d^3 - 109*a^3*b^2*c^2*d^5 + 77*a^4*b*d^7)*x)*sqrt(-b*x^2 + a)*sqrt(d *x + c))/(a^2*b^6*c^4*d^4 - 2*a^3*b^5*c^2*d^6 + a^4*b^4*d^8 + (b^8*c^4*d^4 - 2*a*b^7*c^2*d^6 + a^2*b^6*d^8)*x^4 - 2*(a*b^7*c^4*d^4 - 2*a^2*b^6*c^2*d ^6 + a^3*b^5*d^8)*x^2)
Timed out. \[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x**7/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {x^{7}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^7/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate(x^7/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
\[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {x^{7}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^7/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate(x^7/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^7}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(x^7/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
Output:
int(x^7/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^7}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int(x^7/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
Output:
(231*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt (a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x** 2)*b**2*d**2*x**6),x)*a**6*c*d**4 - 222*sqrt(a - b*x**2)*int(sqrt(c + d*x) /(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x* *2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**5*b*c**3*d**2 - 462*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqr t(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x* *2)*b**2*d**2*x**6),x)*a**5*b*c*d**4*x**2 - 24*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2* sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**4*b**2* c**5 + 444*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*s qrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**4*b**2*c**3*d**2*x**2 + 231*sqrt(a - b*x**2 )*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d* *2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**...