Optimal. Leaf size=241 \[ \frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}-\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.158972, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {459, 279, 321, 329, 275, 217, 206} \[ \frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}-\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 459
Rule 279
Rule 321
Rule 329
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx &=\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac{\left (-15 A b+\frac{9 a B}{2}\right ) \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \, dx}{15 b}\\ &=\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac{(a (10 A b-3 a B)) \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \, dx}{16 b}\\ &=\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac{\left (a^2 (10 A b-3 a B)\right ) \int (e x)^{7/2} \sqrt{a+b x^3} \, dx}{32 b}\\ &=\frac{a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{192 b e}+\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac{\left (a^3 (10 A b-3 a B)\right ) \int \frac{(e x)^{7/2}}{\sqrt{a+b x^3}} \, dx}{128 b}\\ &=\frac{a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{384 b^2}+\frac{a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{192 b e}+\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac{\left (a^4 (10 A b-3 a B) e^3\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{256 b^2}\\ &=\frac{a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{384 b^2}+\frac{a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{192 b e}+\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac{\left (a^4 (10 A b-3 a B) e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{128 b^2}\\ &=\frac{a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{384 b^2}+\frac{a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{192 b e}+\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac{\left (a^4 (10 A b-3 a B) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{384 b^2}\\ &=\frac{a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{384 b^2}+\frac{a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{192 b e}+\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac{\left (a^4 (10 A b-3 a B) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{b x^2}{e^3}} \, dx,x,\frac{(e x)^{3/2}}{\sqrt{a+b x^3}}\right )}{384 b^2}\\ &=\frac{a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt{a+b x^3}}{384 b^2}+\frac{a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt{a+b x^3}}{192 b e}+\frac{a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac{(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac{a^4 (10 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.293084, size = 188, normalized size = 0.78 \[ \frac{e^3 \sqrt{e x} \sqrt{a+b x^3} \left (\sqrt{b} x^{3/2} \sqrt{\frac{b x^3}{a}+1} \left (4 a^2 b^2 x^3 \left (295 A+186 B x^3\right )+30 a^3 b \left (5 A+B x^3\right )-45 a^4 B+16 a b^3 x^6 \left (85 A+63 B x^3\right )+96 b^4 x^9 \left (5 A+4 B x^3\right )\right )+15 a^{7/2} (3 a B-10 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )\right )}{5760 b^{5/2} \sqrt{x} \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.063, size = 8117, normalized size = 33.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 4.58304, size = 940, normalized size = 3.9 \begin{align*} \left [-\frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (384 \, B b^{4} e^{3} x^{13} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{23040 \, b^{2}}, -\frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} b x \sqrt{-\frac{e}{b}}}{2 \, b e x^{3} + a e}\right ) - 2 \,{\left (384 \, B b^{4} e^{3} x^{13} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{11520 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]