Integrand size = 25, antiderivative size = 482 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\frac {4 c \left (5 b c^2-57 a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{315 b^2 d}+\frac {2 \left (10 b c^2-49 a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{315 b^2 d}+\frac {4 c (c+d x)^{5/2} \sqrt {a-b x^2}}{63 b d}-\frac {2 (c+d x)^{7/2} \sqrt {a-b x^2}}{9 b d}+\frac {2 \sqrt {a} \left (10 b^2 c^4-279 a b c^2 d^2-147 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{5/2} d^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} c \left (5 b^2 c^4-62 a b c^2 d^2+57 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 b^{5/2} d^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
4/315*c*(-57*a*d^2+5*b*c^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d+2/315*(-4 9*a*d^2+10*b*c^2)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^2/d+4/63*c*(d*x+c)^(5/2 )*(-b*x^2+a)^(1/2)/b/d-2/9*(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/b/d+2/315*a^(1/2 )*(-147*a^2*d^4-279*a*b*c^2*d^2+10*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^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2) *d))^(1/2)/(-b*x^2+a)^(1/2)-4/315*a^(1/2)*c*(57*a^2*d^4-62*a*b*c^2*d^2+5*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)*El lipticF(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^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.50 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.15 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (\frac {10 b^2 c^4}{d}-279 a b c^2 d-147 a^2 d^3-\frac {b (c+d x) \left (a d^2 (163 c+49 d x)+5 b \left (c^3+15 c^2 d x+19 c d^2 x^2+7 d^3 x^3\right )\right )}{d}+\frac {i b \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (10 b^2 c^4-279 a b c^2 d^2-147 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^3 \left (-a+b x^2\right )}-\frac {i \sqrt {a} \sqrt {b} \left (10 b^2 c^4+155 \sqrt {a} b^{3/2} c^3 d-279 a b c^2 d^2+261 a^{3/2} \sqrt {b} c d^3-147 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{315 b^3 \sqrt {c+d x}} \] Input:
Integrate[(x^2*(c + d*x)^(5/2))/Sqrt[a - b*x^2],x]
Output:
(2*Sqrt[a - b*x^2]*((10*b^2*c^4)/d - 279*a*b*c^2*d - 147*a^2*d^3 - (b*(c + d*x)*(a*d^2*(163*c + 49*d*x) + 5*b*(c^3 + 15*c^2*d*x + 19*c*d^2*x^2 + 7*d ^3*x^3)))/d + (I*b*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(10*b^2*c^4 - 279*a*b*c^ 2*d^2 - 147*a^2*d^4)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sq rt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sq rt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt [b]*c - Sqrt[a]*d)])/(d^3*(-a + b*x^2)) - (I*Sqrt[a]*Sqrt[b]*(10*b^2*c^4 + 155*Sqrt[a]*b^(3/2)*c^3*d - 279*a*b*c^2*d^2 + 261*a^(3/2)*Sqrt[b]*c*d^3 - 147*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)])/(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(315*b^3 *Sqrt[c + d*x])
Time = 0.80 (sec) , antiderivative size = 478, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {604, 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.
| \(\displaystyle \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle -\frac {2 \int -\frac {d (7 a d-2 b c x) (c+d x)^{5/2}}{2 \sqrt {a-b x^2}}dx}{9 b d^2}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(7 a d-2 b c x) (c+d x)^{5/2}}{\sqrt {a-b x^2}}dx}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}-\frac {2 \int -\frac {b (c+d x)^{3/2} \left (39 a c d-\left (10 b c^2-49 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{7 b}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {(c+d x)^{3/2} \left (39 a c d-\left (10 b c^2-49 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}-\frac {2 \int -\frac {3 \sqrt {c+d x} \left (a d \left (55 b c^2+49 a d^2\right )-2 b c \left (5 b c^2-57 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \int \frac {\sqrt {c+d x} \left (a d \left (55 b c^2+49 a d^2\right )-2 b c \left (5 b c^2-57 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )-\frac {2 \int -\frac {b \left (a c d \left (155 b c^2+261 a d^2\right )-\left (10 b^2 c^4-279 a b d^2 c^2-147 a^2 d^4\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \int \frac {a c d \left (155 b c^2+261 a d^2\right )-\left (10 b^2 c^4-279 a b d^2 c^2-147 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 c \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 c \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {2 c \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 c \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 c \sqrt {1-\frac {b x^2}{a}} \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {4 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3 \left (\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-147 a^2 d^4-279 a b c^2 d^2+10 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {4 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (5 b c^2-57 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )+\frac {4}{3} c \sqrt {a-b x^2} \sqrt {c+d x} \left (5 b c^2-57 a d^2\right )\right )}{5 b}+\frac {2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (10 b c^2-49 a d^2\right )}{5 b}\right )+\frac {4}{7} c \sqrt {a-b x^2} (c+d x)^{5/2}}{9 b d}-\frac {2 \sqrt {a-b x^2} (c+d x)^{7/2}}{9 b d}\) |
Input:
Int[(x^2*(c + d*x)^(5/2))/Sqrt[a - b*x^2],x]
Output:
(-2*(c + d*x)^(7/2)*Sqrt[a - b*x^2])/(9*b*d) + ((4*c*(c + d*x)^(5/2)*Sqrt[ a - b*x^2])/7 + ((2*(10*b*c^2 - 49*a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2]) /(5*b) + (3*((4*c*(5*b*c^2 - 57*a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + ((2*Sqrt[a]*(10*b^2*c^4 - 279*a*b*c^2*d^2 - 147*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]) - (4*Sqrt[a]*c*(5*b*c^2 - 57*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)/(9*b*d)
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])
Time = 3.75 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.50
| method | result | size |
| risch | \(-\frac {2 \left (35 b \,d^{3} x^{3}+95 b c \,d^{2} x^{2}+49 a x \,d^{3}+75 b \,c^{2} d x +163 a \,d^{2} c +5 b \,c^{3}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{315 d \,b^{2}}+\frac {\left (\frac {\left (147 a^{2} d^{4}+279 b \,c^{2} d^{2} a -10 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 )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {261 d^{3} c \,a^{2} \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 {155 a \,c^{3} 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}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{315 b^{2} d \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(721\) |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 d^{2} x^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{9 b}-\frac {38 c d \,x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{63 b}-\frac {2 \left (\frac {25 c^{2} d}{21}+\frac {7 a \,d^{3}}{9 b}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (c^{3}-\frac {4 c \left (\frac {25 c^{2} d}{21}+\frac {7 a \,d^{3}}{9 b}\right )}{5 d}+\frac {137 c a \,d^{2}}{63 b}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (\frac {2 a c \left (\frac {25 c^{2} d}{21}+\frac {7 a \,d^{3}}{9 b}\right )}{5 b d}+\frac {a \left (c^{3}-\frac {4 c \left (\frac {25 c^{2} d}{21}+\frac {7 a \,d^{3}}{9 b}\right )}{5 d}+\frac {137 c a \,d^{2}}{63 b}\right )}{3 b}\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 {76 a \,c^{2} d}{63 b}+\frac {3 a \left (\frac {25 c^{2} d}{21}+\frac {7 a \,d^{3}}{9 b}\right )}{5 b}-\frac {2 c \left (c^{3}-\frac {4 c \left (\frac {25 c^{2} d}{21}+\frac {7 a \,d^{3}}{9 b}\right )}{5 d}+\frac {137 c a \,d^{2}}{63 b}\right )}{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}}}\, \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}}\) | \(867\) |
| default | \(\text {Expression too large to display}\) | \(1651\) |
Input:
int(x^2*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/315/d*(35*b*d^3*x^3+95*b*c*d^2*x^2+49*a*d^3*x+75*b*c^2*d*x+163*a*c*d^2+ 5*b*c^3)/b^2*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)+1/315/b^2/d*((147*a^2*d^4+279* a*b*c^2*d^2-10*b^2*c^4)/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)*((c/d-1/b*(a*b)^ (1/2))*EllipticE(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)))+261*d^3*c*a^2/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))+155*a*c^3*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)*Elli pticF(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)))*((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2 )/(-b*x^2+a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.65 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (2 \, {\left (5 \, b^{2} c^{5} + 93 \, a b c^{3} d^{2} + 318 \, a^{2} c d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (10 \, b^{2} c^{4} d - 279 \, a b c^{2} d^{3} - 147 \, a^{2} d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (35 \, b^{2} d^{5} x^{3} + 95 \, b^{2} c d^{4} x^{2} + 5 \, b^{2} c^{3} d^{2} + 163 \, a b c d^{4} + {\left (75 \, b^{2} c^{2} d^{3} + 49 \, a b d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{945 \, b^{3} d^{3}} \] Input:
integrate(x^2*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/945*(2*(5*b^2*c^5 + 93*a*b*c^3*d^2 + 318*a^2*c*d^4)*sqrt(-b*d)*weierstr assPInverse(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*(10*b^2*c^4*d - 279*a*b*c^2*d^3 - 147*a^2*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), -8/2 7*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(35*b^2*d^5*x^3 + 9 5*b^2*c*d^4*x^2 + 5*b^2*c^3*d^2 + 163*a*b*c*d^4 + (75*b^2*c^2*d^3 + 49*a*b *d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*d^3)
\[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {5}{2}}}{\sqrt {a - b x^{2}}}\, dx \] Input:
integrate(x**2*(d*x+c)**(5/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x**2*(c + d*x)**(5/2)/sqrt(a - b*x**2), x)
\[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{2}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate(x^2*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^(5/2)*x^2/sqrt(-b*x^2 + a), x)
\[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{2}} x^{2}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate(x^2*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^(5/2)*x^2/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{5/2}}{\sqrt {a-b\,x^2}} \,d x \] Input:
int((x^2*(c + d*x)^(5/2))/(a - b*x^2)^(1/2),x)
Output:
int((x^2*(c + d*x)^(5/2))/(a - b*x^2)^(1/2), x)
\[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a-b x^2}} \, dx=\frac {-294 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} d^{3}-1210 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,c^{2} d -196 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b c \,d^{2} x -300 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{3} x -380 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{2} d \,x^{2}-140 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c \,d^{2} x^{3}-441 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{4}-837 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{2} d^{2}+30 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c^{4}+147 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} d^{4}+801 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,c^{2} d^{2}+300 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{4}}{630 b^{3} c} \] Input:
int(x^2*(d*x+c)^(5/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 294*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 - 1210*sqrt(c + d*x)*sqrt (a - b*x**2)*a*b*c**2*d - 196*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x - 300*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*x - 380*sqrt(c + d*x)*sqrt( a - b*x**2)*b**2*c**2*d*x**2 - 140*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d **2*x**3 - 441*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 - 837*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 + 30*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 + 147*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 + 801*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**2*b*c**2*d**2 + 300*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 )/(630*b**3*c)