Integrand size = 21, antiderivative size = 457 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {16 \sqrt {-a} \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.31 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 829, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {16 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {16 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (33 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {8 c \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{63 e^5}-\frac {20 c \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]
[In]
[Out]
Rule 430
Rule 435
Rule 733
Rule 747
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {(10 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e} \\ & = -\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {40 \int \frac {\left (-\frac {1}{2} a c d e+\frac {1}{2} c \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{21 e^3} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {32 \int \frac {-a c^2 d e \left (2 c d^2+3 a e^2\right )+\frac {1}{4} c^2 \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{63 c e^5} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (8 c d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{63 e^6}+\frac {\left (8 c \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{63 e^6} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (16 a \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{63 \sqrt {-a} e^6 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (16 a \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{63 \sqrt {-a} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {16 \sqrt {-a} \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 25.27 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (63 a^2 e^4+2 a c e^2 \left (106 d^2+29 d e x-14 e^2 x^2\right )+c^2 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )}{e^5 (d+e x)}+\frac {16 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{5/2} d^5+32 \sqrt {a} c^2 d^4 e-57 i a c^{3/2} d^3 e^2+57 a^{3/2} c d^2 e^3-21 i a^2 \sqrt {c} d e^4+21 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c^2 d^4+8 i \sqrt {a} c^{3/2} d^3 e+57 a c d^2 e^2+12 i a^{3/2} \sqrt {c} d e^3+21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^7 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{63 \sqrt {a+c x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1262\) vs. \(2(381)=762\).
Time = 4.90 (sec) , antiderivative size = 1263, normalized size of antiderivative = 2.76
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1263\) |
default | \(\text {Expression too large to display}\) | \(1736\) |
risch | \(\text {Expression too large to display}\) | \(1745\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (32 \, c^{2} d^{6} + 81 \, a c d^{4} e^{2} + 57 \, a^{2} d^{2} e^{4} + {\left (32 \, c^{2} d^{5} e + 81 \, a c d^{3} e^{3} + 57 \, a^{2} d e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{2} d^{5} e + 57 \, a c d^{3} e^{3} + 21 \, a^{2} d e^{5} + {\left (32 \, c^{2} d^{4} e^{2} + 57 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (7 \, c^{2} e^{6} x^{4} - 10 \, c^{2} d e^{5} x^{3} - 128 \, c^{2} d^{4} e^{2} - 212 \, a c d^{2} e^{4} - 63 \, a^{2} e^{6} + 4 \, {\left (4 \, c^{2} d^{2} e^{4} + 7 \, a c e^{6}\right )} x^{2} - 2 \, {\left (16 \, c^{2} d^{3} e^{3} + 29 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{189 \, {\left (e^{8} x + d e^{7}\right )}} \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
[In]
[Out]