Integrand size = 21, antiderivative size = 494 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {16 \sqrt {-a} \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.34 (sec) , antiderivative size = 494, 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 = {749, 829, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=-\frac {16 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 \sqrt {c} e^6 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {8 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{693 e^5}+\frac {20 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{693 e^3}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e} \]
[In]
[Out]
Rule 430
Rule 435
Rule 733
Rule 749
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {10 \int \frac {(a e-c d x) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 e} \\ & = \frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {40 \int \frac {\left (\frac {1}{2} a c e \left (c d^2+9 a e^2\right )-4 c^2 d \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{231 c e^3} \\ & = \frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {32 \int \frac {\frac {1}{4} a c^2 e \left (8 c^2 d^4+21 a c d^2 e^2+45 a^2 e^4\right )-\frac {1}{4} c^3 d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{693 c^2 e^5} \\ & = \frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {\left (8 \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{693 e^6}-\frac {\left (8 c d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{693 e^6} \\ & = \frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}-\frac {\left (16 a \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 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 )}{693 \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 \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\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 )}{693 \sqrt {-a} \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {16 \sqrt {-a} \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 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 )}{693 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\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 )}{693 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 24.96 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (-\frac {8 d e^2 \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \left (a+c x^2\right )}{d+e x}+e^2 \left (a+c x^2\right ) \left (333 a^2 e^4+2 a c e^2 \left (178 d^2-131 d e x+108 e^2 x^2\right )+c^2 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )-8 i c d \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\frac {8 \sqrt {a} e \left (32 c^{5/2} d^5+8 i \sqrt {a} c^2 d^4 e+93 a c^{3/2} d^3 e^2+21 i a^{3/2} c d^2 e^3+93 a^2 \sqrt {c} d e^4+45 i a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{693 e^7 \sqrt {a+c x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1147\) vs. \(2(416)=832\).
Time = 3.22 (sec) , antiderivative size = 1148, normalized size of antiderivative = 2.32
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1148\) |
elliptic | \(\text {Expression too large to display}\) | \(1151\) |
default | \(\text {Expression too large to display}\) | \(1970\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{6} + 117 \, a c^{2} d^{4} e^{2} + 156 \, a^{2} c d^{2} e^{4} + 135 \, a^{3} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{3} d^{5} e + 93 \, a c^{2} d^{3} e^{3} + 93 \, a^{2} c d e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (63 \, c^{3} e^{6} x^{4} - 70 \, c^{3} d e^{5} x^{3} + 128 \, c^{3} d^{4} e^{2} + 356 \, a c^{2} d^{2} e^{4} + 333 \, a^{2} c e^{6} + 8 \, {\left (10 \, c^{3} d^{2} e^{4} + 27 \, a c^{2} e^{6}\right )} x^{2} - 2 \, {\left (48 \, c^{3} d^{3} e^{3} + 131 \, a c^{2} d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{2079 \, c e^{7}} \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\sqrt {d + e x}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {e x + d}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {e x + d}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{\sqrt {d+e\,x}} \,d x \]
[In]
[Out]