Integrand size = 25, antiderivative size = 554 \[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=-\frac {2 \left (8 b^2 c^4+57 a b c^2 d^2+135 a^2 d^4\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{693 b^3 d^2}-\frac {2 c \left (8 b c^2+67 a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{693 b^2 d^2}-\frac {2 \left (8 b c^2+81 a d^2\right ) (c+d x)^{5/2} \sqrt {a-b x^2}}{693 b^2 d^2}+\frac {26 c (c+d x)^{7/2} \sqrt {a-b x^2}}{99 b d^2}-\frac {2 (c+d x)^{9/2} \sqrt {a-b x^2}}{11 b d^2}-\frac {2 \sqrt {a} c \left (8 b^2 c^4+51 a b c^2 d^2+741 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{693 b^{5/2} d^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (8 b^3 c^6+49 a b^2 c^4 d^2+78 a^2 b c^2 d^4-135 a^3 d^6\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{693 b^{7/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/693*(135*a^2*d^4+57*a*b*c^2*d^2+8*b^2*c^4)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/ 2)/b^3/d^2-2/693*c*(67*a*d^2+8*b*c^2)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^2/d ^2-2/693*(81*a*d^2+8*b*c^2)*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b^2/d^2+26/99*c *(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/b/d^2-2/11*(d*x+c)^(9/2)*(-b*x^2+a)^(1/2)/ b/d^2-2/693*a^(1/2)*c*(741*a^2*d^4+51*a*b*c^2*d^2+8*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^(5/2)/d^3/(b^(1/2)*(d*x+c)/( b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+2/693*a^(1/2)*(-135*a^3*d^6+7 8*a^2*b*c^2*d^4+49*a*b^2*c^4*d^2+8*b^3*c^6)*(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/(d* x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.60 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-c \left (8 b^2 c^4+51 a b c^2 d^2+741 a^2 d^4\right )+(c+d x) \left (-135 a^2 d^4-a b d^2 \left (205 c^2+229 c d x+81 d^2 x^2\right )+b^2 \left (4 c^4-3 c^3 d x-113 c^2 d^2 x^2-161 c d^3 x^3-63 d^4 x^4\right )\right )-\frac {i b c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (8 b^2 c^4+51 a b c^2 d^2+741 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \left (-a+b x^2\right )}+\frac {i \sqrt {a} \left (8 b^{5/2} c^5-2 \sqrt {a} b^2 c^4 d+51 a b^{3/2} c^3 d^2-663 a^{3/2} b c^2 d^3+741 a^2 \sqrt {b} c d^4-135 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{693 b^3 d^2 \sqrt {c+d x}} \] Input:
Integrate[(x^3*(c + d*x)^(5/2))/Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*(-(c*(8*b^2*c^4 + 51*a*b*c^2*d^2 + 741*a^2*d^4)) + (c + d*x)*(-135*a^2*d^4 - a*b*d^2*(205*c^2 + 229*c*d*x + 81*d^2*x^2) + b^2*(4* c^4 - 3*c^3*d*x - 113*c^2*d^2*x^2 - 161*c*d^3*x^3 - 63*d^4*x^4)) - (I*b*c* Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(8*b^2*c^4 + 51*a*b*c^2*d^2 + 741*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)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/S qrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/ (d^2*(-a + b*x^2)) + (I*Sqrt[a]*(8*b^(5/2)*c^5 - 2*Sqrt[a]*b^2*c^4*d + 51* a*b^(3/2)*c^3*d^2 - 663*a^(3/2)*b*c^2*d^3 + 741*a^2*Sqrt[b]*c*d^4 - 135*a^ (5/2)*d^5)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/S qrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (S qrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sq rt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(693*b^3*d^2* Sqrt[c + d*x])
Time = 1.05 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {604, 27, 2185, 27, 687, 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.
| \(\displaystyle \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle -\frac {2 \int -\frac {(c+d x)^{5/2} \left (-13 b c x^2 d^2+9 a c d^2-\left (2 b c^2-9 a d^2\right ) x d\right )}{2 \sqrt {a-b x^2}}dx}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (-13 b c x^2 d^2+9 a c d^2-\left (2 b c^2-9 a d^2\right ) x d\right )}{\sqrt {a-b x^2}}dx}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {2 \int \frac {b d^3 (c+d x)^{5/2} \left (10 a c d-\left (8 b c^2+81 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{9 b d^2}}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \int \frac {(c+d x)^{5/2} \left (10 a c d-\left (8 b c^2+81 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}-\frac {2 \int -\frac {5 (c+d x)^{3/2} \left (3 a d \left (2 b c^2-27 a d^2\right )-b c \left (8 b c^2+67 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \int \frac {(c+d x)^{3/2} \left (3 a d \left (2 b c^2-27 a d^2\right )-b c \left (8 b c^2+67 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )-\frac {2 \int -\frac {3 b \sqrt {c+d x} \left (2 a c d \left (b c^2-101 a d^2\right )-\left (8 b^2 c^4+57 a b d^2 c^2+135 a^2 d^4\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \int \frac {\sqrt {c+d x} \left (2 a c d \left (b c^2-101 a d^2\right )-\left (8 b^2 c^4+57 a b d^2 c^2+135 a^2 d^4\right ) x\right )}{\sqrt {a-b x^2}}dx+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {2 \int \frac {a d \left (2 b^2 c^4+663 a b d^2 c^2+135 a^2 d^4\right )+b c \left (8 b^2 c^4+51 a b d^2 c^2+741 a^2 d^4\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {\int \frac {a d \left (2 b^2 c^4+663 a b d^2 c^2+135 a^2 d^4\right )+b c \left (8 b^2 c^4+51 a b d^2 c^2+741 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {\frac {b c \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {\frac {b c \sqrt {1-\frac {b x^2}{a}} \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {-\frac {\left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {26}{9} c d \sqrt {a-b x^2} (c+d x)^{7/2}-\frac {1}{9} d \left (\frac {5 \left (\frac {3}{5} \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right )}{3 b}-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (135 a^2 d^4+57 a b c^2 d^2+8 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (741 a^2 d^4+51 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}\right )+\frac {2}{5} c \sqrt {a-b x^2} (c+d x)^{3/2} \left (67 a d^2+8 b c^2\right )\right )}{7 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{5/2} \left (81 a d^2+8 b c^2\right )}{7 b}\right )}{11 b d^3}-\frac {2 \sqrt {a-b x^2} (c+d x)^{9/2}}{11 b d^2}\) |
Input:
Int[(x^3*(c + d*x)^(5/2))/Sqrt[a - b*x^2],x]
Output:
(-2*(c + d*x)^(9/2)*Sqrt[a - b*x^2])/(11*b*d^2) + ((26*c*d*(c + d*x)^(7/2) *Sqrt[a - b*x^2])/9 - (d*((2*(8*b*c^2 + 81*a*d^2)*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b) + (5*((2*c*(8*b*c^2 + 67*a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b* x^2])/5 + (3*((2*(8*b^2*c^4 + 57*a*b*c^2*d^2 + 135*a^2*d^4)*Sqrt[c + d*x]* Sqrt[a - b*x^2])/(3*b) - ((-2*Sqrt[a]*Sqrt[b]*c*(8*b^2*c^4 + 51*a*b*c^2*d^ 2 + 741*a^2*d^4)*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)])/(d*Sqr t[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[ a]*(b*c^2 - a*d^2)*(8*b^2*c^4 + 57*a*b*c^2*d^2 + 135*a^2*d^4)*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*b)))/5))/(7*b)))/9)/(11*b*d ^3)
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_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(b*d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b* x^2)^p*ExpandToSum[b*d^m*(m + n + 2*p + 1)*x^m - b*(m + n + 2*p + 1)*(c + d *x)^m - (c + d*x)^(m - 2)*(a*d^2*(m + n - 1) - b*c^2*(m + n + 2*p + 1) - 2* b*c*d*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && IntegerQ[2*p]
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]))
Time = 4.53 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(-\frac {2 \left (63 b^{2} d^{4} x^{4}+161 b^{2} c \,d^{3} x^{3}+81 a b \,d^{4} x^{2}+113 d^{2} c^{2} x^{2} b^{2}+229 a b c \,d^{3} x +3 b^{2} c^{3} d x +135 a^{2} d^{4}+205 b \,c^{2} d^{2} a -4 b^{2} c^{4}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{693 d^{2} b^{3}}+\frac {\left (\frac {c \left (741 a^{2} d^{4}+51 b \,c^{2} d^{2} a +8 b^{2} c^{4}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {135 a^{3} d^{5} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 a b \,c^{4} d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {663 a^{2} c^{2} d^{3} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{693 b^{3} d^{2} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(933\) |
| elliptic | \(\text {Expression too large to display}\) | \(1111\) |
| default | \(\text {Expression too large to display}\) | \(1880\) |
Input:
int(x^3*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/693/d^2*(63*b^2*d^4*x^4+161*b^2*c*d^3*x^3+81*a*b*d^4*x^2+113*b^2*c^2*d^ 2*x^2+229*a*b*c*d^3*x+3*b^2*c^3*d*x+135*a^2*d^4+205*a*b*c^2*d^2-4*b^2*c^4) /b^3*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)+1/693/b^3/d^2*(c*(741*a^2*d^4+51*a*b*c ^2*d^2+8*b^2*c^4)*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^ (1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*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))*E llipticE(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b) ^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*EllipticF(1/2*2^(1/2)*((x+1/b*(a* b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1 /2)))+135*a^3*d^5/b*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2) )^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a *b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)* ((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b) ^(1/2)))^(1/2))+2*a*b*c^4*d*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a* b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/ 2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2* 2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1 /b*(a*b)^(1/2)))^(1/2))+663*a^2*c^2*d^3*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^ (1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1 /b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)...
Time = 0.09 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.67 \[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, b^{3} c^{6} + 45 \, a b^{2} c^{4} d^{2} - 1248 \, a^{2} b c^{2} d^{4} - 405 \, a^{3} d^{6}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, b^{3} c^{5} d + 51 \, a b^{2} c^{3} d^{3} + 741 \, a^{2} b c d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (63 \, b^{3} d^{6} x^{4} + 161 \, b^{3} c d^{5} x^{3} - 4 \, b^{3} c^{4} d^{2} + 205 \, a b^{2} c^{2} d^{4} + 135 \, a^{2} b d^{6} + {\left (113 \, b^{3} c^{2} d^{4} + 81 \, a b^{2} d^{6}\right )} x^{2} + {\left (3 \, b^{3} c^{3} d^{3} + 229 \, a b^{2} c d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{2079 \, b^{4} d^{4}} \] Input:
integrate(x^3*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/2079*((8*b^3*c^6 + 45*a*b^2*c^4*d^2 - 1248*a^2*b*c^2*d^4 - 405*a^3*d^6)* 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*(8*b^3*c^5*d + 51*a*b^2*c^3* d^3 + 741*a^2*b*c*d^5)*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*(63*b^3*d^6*x^4 + 161*b^3*c*d^5*x^3 - 4*b^3*c^4*d^2 + 205*a*b^2*c^2*d^ 4 + 135*a^2*b*d^6 + (113*b^3*c^2*d^4 + 81*a*b^2*d^6)*x^2 + (3*b^3*c^3*d^3 + 229*a*b^2*c*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^4*d^4)
\[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int \frac {x^{3} \left (c + d x\right )^{\frac {5}{2}}}{\sqrt {a - b x^{2}}}\, dx \] Input:
integrate(x**3*(d*x+c)**(5/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x**3*(c + d*x)**(5/2)/sqrt(a - b*x**2), x)
\[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{3}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate(x^3*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)*x^3/sqrt(-b*x^2 + a), x)
\[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{3}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate(x^3*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)*x^3/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int \frac {x^3\,{\left (c+d\,x\right )}^{5/2}}{\sqrt {a-b\,x^2}} \,d x \] Input:
int((x^3*(c + d*x)^(5/2))/(a - b*x^2)^(1/2),x)
Output:
int((x^3*(c + d*x)^(5/2))/(a - b*x^2)^(1/2), x)
\[ \int \frac {x^3 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\frac {-2022 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} d^{3}-922 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,c^{2} d -916 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b c \,d^{2} x -324 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,d^{3} x^{2}-12 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{3} x -452 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{2} d \,x^{2}-644 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c \,d^{2} x^{3}-252 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} d^{3} x^{4}-2223 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{4}-153 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{2} d^{2}-24 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c^{4}+1011 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} d^{4}+1377 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,c^{2} d^{2}+12 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{4}}{1386 b^{3} d} \] Input:
int(x^3*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 2022*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 - 922*sqrt(c + d*x)*sqrt (a - b*x**2)*a*b*c**2*d - 916*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x - 324*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*d**3*x**2 - 12*sqrt(c + d*x)*sqrt (a - b*x**2)*b**2*c**3*x - 452*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d* x**2 - 644*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**2*x**3 - 252*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*d**3*x**4 - 2223*int((sqrt(c + d*x)*sqrt(a - b* x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*d**4 - 153*int(( sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x )*a*b**2*c**2*d**2 - 24*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a *d*x - b*c*x**2 - b*d*x**3),x)*b**3*c**4 + 1011*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**3*d**4 + 1377*int((sq rt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2* b*c**2*d**2 + 12*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x **2 - b*d*x**3),x)*a*b**2*c**4)/(1386*b**3*d)