Integrand size = 25, antiderivative size = 505 \[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {a x (c+d x)^{5/2}}{b^2 \sqrt {a-b x^2}}-\frac {c \left (\frac {20 b c^2}{d}-879 a d\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{315 b^3}-\frac {\left (\frac {20 b c^2}{d}-539 a d\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{315 b^3}-\frac {4 c (c+d x)^{5/2} \sqrt {a-b x^2}}{63 b^2 d}+\frac {2 (c+d x)^{7/2} \sqrt {a-b x^2}}{9 b^2 d}-\frac {\sqrt {a} \left (20 b^2 c^4-1839 a b c^2 d^2-1617 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{7/2} d^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {a} c \left (20 b^2 c^4-899 a b c^2 d^2+879 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{7/2} d^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
a*x*(d*x+c)^(5/2)/b^2/(-b*x^2+a)^(1/2)-1/315*c*(20*b*c^2/d-879*a*d)*(d*x+c )^(1/2)*(-b*x^2+a)^(1/2)/b^3-1/315*(20*b*c^2/d-539*a*d)*(d*x+c)^(3/2)*(-b* x^2+a)^(1/2)/b^3-4/63*c*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b^2/d+2/9*(d*x+c)^( 7/2)*(-b*x^2+a)^(1/2)/b^2/d-1/315*a^(1/2)*(-1617*a^2*d^4-1839*a*b*c^2*d^2+ 20*b^2*c^4)*(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^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+1/31 5*a^(1/2)*c*(879*a^2*d^4-899*a*b*c^2*d^2+20*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^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.15 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-a^2 d^2 (1418 c+539 d x)+10 b^2 x^2 \left (c^3+15 c^2 d x+19 c d^2 x^2+7 d^3 x^3\right )+a b \left (-10 c^3-465 c^2 d x+598 c d^2 x^2+154 d^3 x^3\right )\right )}{b^3 d \left (-a+b x^2\right )}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (20 b^2 c^4-1839 a b c^2 d^2-1617 a^2 d^4\right ) \left (-a+b x^2\right )-i \sqrt {b} \left (20 b^{5/2} c^5-20 \sqrt {a} b^2 c^4 d-1839 a b^{3/2} c^3 d^2+1839 a^{3/2} b c^2 d^3-1617 a^2 \sqrt {b} c d^4+1617 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \sqrt {b} d \left (20 b^2 c^4+940 \sqrt {a} b^{3/2} c^3 d-1839 a b c^2 d^2+2496 a^{3/2} \sqrt {b} c d^3-1617 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b^4 d^3 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{315 \sqrt {c+d x}} \] Input:
Integrate[(x^4*(c + d*x)^(5/2))/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(-(a^2*d^2*(1418*c + 539*d*x)) + 10*b^2*x^2*( c^3 + 15*c^2*d*x + 19*c*d^2*x^2 + 7*d^3*x^3) + a*b*(-10*c^3 - 465*c^2*d*x + 598*c*d^2*x^2 + 154*d^3*x^3)))/(b^3*d*(-a + b*x^2)) - (d^2*Sqrt[-c + (Sq rt[a]*d)/Sqrt[b]]*(20*b^2*c^4 - 1839*a*b*c^2*d^2 - 1617*a^2*d^4)*(-a + b*x ^2) - I*Sqrt[b]*(20*b^(5/2)*c^5 - 20*Sqrt[a]*b^2*c^4*d - 1839*a*b^(3/2)*c^ 3*d^2 + 1839*a^(3/2)*b*c^2*d^3 - 1617*a^2*Sqrt[b]*c*d^4 + 1617*a^(5/2)*d^5 )*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*(20*b^2*c^4 + 940*Sqrt[a]*b^(3/2)*c^3*d - 1839*a*b* c^2*d^2 + 2496*a^(3/2)*Sqrt[b]*c*d^3 - 1617*a^2*d^4)*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^3*Sqrt[-c + (S qrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(315*Sqrt[c + d*x])
Time = 1.18 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.18, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {602, 27, 2185, 27, 687, 27, 27, 687, 27, 687, 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.
| \(\displaystyle \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 602 |
| \(\displaystyle \frac {\int -\frac {(c+d x)^{5/2} \left (\frac {\left (2 b c^2-7 a d^2\right ) a^2}{b^2}+\frac {7 c d x a^2}{b}+2 \left (c^2-\frac {a d^2}{b}\right ) x^2 a\right )}{2 \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(c+d x)^{5/2} \left (\frac {\left (2 b c^2-7 a d^2\right ) a^2}{b^2}+\frac {7 c d x a^2}{b}+2 \left (c^2-\frac {a d^2}{b}\right ) x^2 a\right )}{\sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {-\frac {2 \int -\frac {a d (c+d x)^{5/2} \left (a d \left (32 c^2-\frac {77 a d^2}{b}\right )-c \left (4 b c^2-67 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{9 b d^2}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \int \frac {(c+d x)^{5/2} \left (a d \left (32 c^2-\frac {77 a d^2}{b}\right )-c \left (4 b c^2-67 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {\frac {a \left (\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}-\frac {2 \int -\frac {(c+d x)^{3/2} \left (204 a c d \left (b c^2-a d^2\right )-\left (20 b c^2-539 a d^2\right ) \left (b c^2-a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\int \frac {\left (b c^2-a d^2\right ) (c+d x)^{3/2} \left (204 a c d-\left (20 b c^2-539 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \int \frac {(c+d x)^{3/2} \left (204 a c d-\left (20 b c^2-539 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}-\frac {2 \int -\frac {3 \sqrt {c+d x} \left (a d \left (320 b c^2+539 a d^2\right )-b c \left (20 b c^2-879 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \int \frac {\sqrt {c+d x} \left (a d \left (320 b c^2+539 a d^2\right )-b c \left (20 b c^2-879 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )-\frac {2 \int -\frac {b \left (4 a c d \left (235 b c^2+624 a d^2\right )-\left (20 b^2 c^4-1839 a b d^2 c^2-1617 a^2 d^4\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \int \frac {4 a c d \left (235 b c^2+624 a d^2\right )-\left (20 b^2 c^4-1839 a b d^2 c^2-1617 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {c \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {c \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {c \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {c \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {c \sqrt {1-\frac {b x^2}{a}} \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-1617 a^2 d^4-1839 a b c^2 d^2+20 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (20 b c^2-879 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )+\frac {2}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (20 b c^2-879 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (20 b c^2-539 a d^2\right )}{5 b}\right )}{7 b}+\frac {2 c \sqrt {a-b x^2} (c+d x)^{5/2} \left (4 b c^2-67 a d^2\right )}{7 b}\right )}{9 b d}-\frac {4 a \sqrt {a-b x^2} (c+d x)^{7/2} \left (b c^2-a d^2\right )}{9 b^2 d}}{2 a \left (b c^2-a d^2\right )}-\frac {a (c+d x)^{7/2} (a d-b c x)}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\) |
Input:
Int[(x^4*(c + d*x)^(5/2))/(a - b*x^2)^(3/2),x]
Output:
-((a*(a*d - b*c*x)*(c + d*x)^(7/2))/(b^2*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - ((-4*a*(b*c^2 - a*d^2)*(c + d*x)^(7/2)*Sqrt[a - b*x^2])/(9*b^2*d) + (a* ((2*c*(4*b*c^2 - 67*a*d^2)*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b) + ((b*c^ 2 - a*d^2)*((2*(20*b*c^2 - 539*a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5* b) + (3*((2*c*(20*b*c^2 - 879*a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + (( 2*Sqrt[a]*(20*b^2*c^4 - 1839*a*b*c^2*d^2 - 1617*a^2*d^4)*Sqrt[c + d*x]*Sqr t[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))/(Sqr t[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*c*(20*b*c^2 - 879*a*d^2 )*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[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)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/ 3))/(5*b)))/(7*b)))/(9*b*d))/(2*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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1167\) vs. \(2(419)=838\).
Time = 9.45 (sec) , antiderivative size = 1168, normalized size of antiderivative = 2.31
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1168\) |
| risch | \(\text {Expression too large to display}\) | \(1251\) |
| default | \(\text {Expression too large to display}\) | \(1651\) |
Input:
int(x^4*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x-b*c) *(1/2*(a*d^2+b*c^2)*a/b^4*x+a^2*c*d/b^4)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2/ 9*d^2/b^2*x^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+38/63*c*d/b^2*x^2*(-b*d*x ^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(-d*(a*d^2+3*b*c^2)/b^2-7/9*d^3/b^2*a+38/2 1*c^2*d/b)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(-(3*a*d^2+b*c^2)/ b^2*c-137/63*d^2/b^2*a*c-4/5*(-d*(a*d^2+3*b*c^2)/b^2-7/9*d^3/b^2*a+38/21*c ^2*d/b)/d*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-1/b^3*d^2*a^2*c-1/ b^3*c*(a*d^2+b*c^2)*a+2/5*(-d*(a*d^2+3*b*c^2)/b^2-7/9*d^3/b^2*a+38/21*c^2* d/b)/b/d*a*c+1/3*(-(3*a*d^2+b*c^2)/b^2*c-137/63*d^2/b^2*a*c-4/5*(-d*(a*d^2 +3*b*c^2)/b^2-7/9*d^3/b^2*a+38/21*c^2*d/b)/d*c)/b*a)*(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*(-a*d*(a*d^2+3* b*c^2)/b^3-1/2*(a*d^2+b*c^2)*a*d/b^3-76/63*c^2*d/b^2*a+3/5*(-d*(a*d^2+3*b* c^2)/b^2-7/9*d^3/b^2*a+38/21*c^2*d/b)/b*a-2/3*(-(3*a*d^2+b*c^2)/b^2*c-137/ 63*d^2/b^2*a*c-4/5*(-d*(a*d^2+3*b*c^2)/b^2-7/9*d^3/b^2*a+38/21*c^2*d/b)/d* c)/d*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)*((-c/d-1/b*(a*...
Time = 0.10 (sec) , antiderivative size = 465, normalized size of antiderivative = 0.92 \[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=-\frac {{\left (20 \, a b^{2} c^{5} + 981 \, a^{2} b c^{3} d^{2} + 5871 \, a^{3} c d^{4} - {\left (20 \, b^{3} c^{5} + 981 \, a b^{2} c^{3} d^{2} + 5871 \, a^{2} b c d^{4}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (20 \, a b^{2} c^{4} d - 1839 \, a^{2} b c^{2} d^{3} - 1617 \, a^{3} d^{5} - {\left (20 \, b^{3} c^{4} d - 1839 \, a b^{2} c^{2} d^{3} - 1617 \, a^{2} b d^{5}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (70 \, b^{3} d^{5} x^{5} + 190 \, b^{3} c d^{4} x^{4} - 10 \, a b^{2} c^{3} d^{2} - 1418 \, a^{2} b c d^{4} + 2 \, {\left (75 \, b^{3} c^{2} d^{3} + 77 \, a b^{2} d^{5}\right )} x^{3} + 2 \, {\left (5 \, b^{3} c^{3} d^{2} + 299 \, a b^{2} c d^{4}\right )} x^{2} - {\left (465 \, a b^{2} c^{2} d^{3} + 539 \, a^{2} b d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{945 \, {\left (b^{5} d^{3} x^{2} - a b^{4} d^{3}\right )}} \] Input:
integrate(x^4*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/945*((20*a*b^2*c^5 + 981*a^2*b*c^3*d^2 + 5871*a^3*c*d^4 - (20*b^3*c^5 + 981*a*b^2*c^3*d^2 + 5871*a^2*b*c*d^4)*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*(20*a*b^2*c^4*d - 1839*a^2*b*c^2*d^3 - 1617*a^3*d^5 - (20* b^3*c^4*d - 1839*a*b^2*c^2*d^3 - 1617*a^2*b*d^5)*x^2)*sqrt(-b*d)*weierstra ssZeta(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), w eierstrassPInverse(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*(70*b^3*d^5*x^5 + 190*b^3*c*d^4*x^4 - 1 0*a*b^2*c^3*d^2 - 1418*a^2*b*c*d^4 + 2*(75*b^3*c^2*d^3 + 77*a*b^2*d^5)*x^3 + 2*(5*b^3*c^3*d^2 + 299*a*b^2*c*d^4)*x^2 - (465*a*b^2*c^2*d^3 + 539*a^2* b*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^5*d^3*x^2 - a*b^4*d^3)
Timed out. \[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x**4*(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)*x^4/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)*x^4/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (c+d\,x\right )}^{5/2}}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^4*(c + d*x)^(5/2))/(a - b*x^2)^(3/2),x)
Output:
int((x^4*(c + d*x)^(5/2))/(a - b*x^2)^(3/2), x)
\[ \int \frac {x^4 (c+d x)^{5/2}}{\left (a-b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int(x^4*(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
( - 1617*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqr t(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2 )*b*d**2*x**4),x)*a**5*c*d**6 - 222*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sq rt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b* c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**4*b*c**3*d**4 + 1617*sqrt( a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)* a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4) ,x)*a**4*b*c*d**6*x**2 + 1859*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x **2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**3*b**2*c**5*d**2 + 222*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d* *2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)* a**3*b**2*c**3*d**4*x**2 - 20*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x **2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b**3*c**7 - 1859*sqrt(a - b*x* *2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x **2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**2 *b**3*c**5*d**2*x**2 + 20*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x **2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a*b**4*c**7*x**2 - 539*sqrt(a - b*x*...