Optimal. Leaf size=120 \[ -\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0864786, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {459, 288, 329, 275, 217, 206} \[ -\frac{e^2 (e x)^{3/2} (2 A b-3 a B)}{3 b^2 \sqrt{a+b x^3}}+\frac{e^{7/2} (2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{5/2}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 459
Rule 288
Rule 329
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(e x)^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}-\frac{\left (-3 A b+\frac{9 a B}{2}\right ) \int \frac{(e x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}+\frac{\left ((2 A b-3 a B) e^3\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^3}} \, dx}{2 b^2}\\ &=-\frac{(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}+\frac{\left ((2 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 )}{b^2}\\ &=-\frac{(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}+\frac{\left ((2 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 )}{3 b^2}\\ &=-\frac{(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}+\frac{\left ((2 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 )}{3 b^2}\\ &=-\frac{(2 A b-3 a B) e^2 (e x)^{3/2}}{3 b^2 \sqrt{a+b x^3}}+\frac{B (e x)^{9/2}}{3 b e \sqrt{a+b x^3}}+\frac{(2 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 )}{3 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.163377, size = 109, normalized size = 0.91 \[ \frac{e^3 \sqrt{e x} \left (\sqrt{b} x^{3/2} \left (3 a B-2 A b+b B x^3\right )-\sqrt{a} \sqrt{\frac{b x^3}{a}+1} (3 a B-2 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )\right )}{3 b^{5/2} \sqrt{x} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.065, size = 7016, normalized size = 58.5 \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 \frac{{\left (B x^{3} + A\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 4.33937, size = 667, normalized size = 5.56 \begin{align*} \left [-\frac{{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} \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 (B b e^{3} x^{4} +{\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{12 \,{\left (b^{3} x^{3} + a b^{2}\right )}}, \frac{{\left ({\left (3 \, B a b - 2 \, A b^{2}\right )} e^{3} x^{3} +{\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} \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 (B b e^{3} x^{4} +{\left (3 \, B a - 2 \, A b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )}}\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(-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]