Integrand size = 21, antiderivative size = 497 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \sqrt {-a} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 \sqrt {c} e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.39 (sec) , antiderivative size = 497, 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 = {757, 829, 858, 733, 435, 430} \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=-\frac {8 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}+\frac {32 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 \sqrt {c} e^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c} \]
[In]
[Out]
Rule 430
Rule 435
Rule 733
Rule 757
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {2 \int \frac {\left (\frac {1}{2} \left (11 c d^2-a e^2\right )+6 c d e x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 c} \\ & = \frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {8 \int \frac {\left (\frac {3}{4} a c e^2 \left (29 c d^2-3 a e^2\right )-\frac {3}{4} c^2 d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{231 c^2 e^2} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \int \frac {\frac {3}{8} a c^2 e^2 \left (c^2 d^4+114 a c d^2 e^2-15 a^2 e^4\right )-\frac {3}{2} c^3 d e \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3465 c^3 e^4} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {\left (16 d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{1155 e^4}+\frac {\left (4 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{1155 c e^4} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {\left (32 a d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\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 )}{1155 \sqrt {-a} \sqrt {c} e^4 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (8 a \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 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 )}{1155 \sqrt {-a} c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \sqrt {-a} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\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 )}{1155 \sqrt {c} e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 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 )}{1155 c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 25.71 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.40 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (60 a^2 e^4+a c e^2 \left (47 d^2+326 d e x+195 e^2 x^2\right )+c^2 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )}{c e^3}-\frac {8 \left (4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c^2 d^4+6 a c d^2 e^2-27 a^2 e^4\right ) \left (a+c x^2\right )+4 \sqrt {c} d \left (-i c^{5/2} d^5+\sqrt {a} c^2 d^4 e-6 i a c^{3/2} d^3 e^2+6 a^{3/2} c d^2 e^3+27 i a^2 \sqrt {c} d e^4-27 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (-4 c^{5/2} d^5-i \sqrt {a} c^2 d^4 e-24 a c^{3/2} d^3 e^2-114 i a^{3/2} c d^2 e^3+108 a^2 \sqrt {c} d e^4+15 i a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{1155 \sqrt {a+c x^2}} \]
[In]
[Out]
Timed out.
hanged
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.74 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (4 \, {\left (4 \, c^{3} d^{6} + 27 \, a c^{2} d^{4} e^{2} + 234 \, a^{2} c d^{2} e^{4} - 45 \, a^{3} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 48 \, {\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} - 27 \, a^{2} c d e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (105 \, c^{3} e^{6} x^{4} + 140 \, c^{3} d e^{5} x^{3} + 8 \, c^{3} d^{4} e^{2} + 47 \, a c^{2} d^{2} e^{4} + 60 \, a^{2} c e^{6} + 5 \, {\left (c^{3} d^{2} e^{4} + 39 \, a c^{2} e^{6}\right )} x^{2} - 2 \, {\left (3 \, c^{3} d^{3} e^{3} - 163 \, a c^{2} d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{3465 \, c^{2} e^{5}} \]
[In]
[Out]
\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
[In]
[Out]
\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
[In]
[Out]