Optimal. Leaf size=169 \[ -\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{2 b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{3 \sqrt{d} e^2 \sqrt{-c^2 x^2}}-\frac{b c x \sqrt{-c^2 x^2-1}}{3 e \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.268162, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {266, 43, 6302, 12, 573, 152, 93, 204} \[ -\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{2 b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{3 \sqrt{d} e^2 \sqrt{-c^2 x^2}}-\frac{b c x \sqrt{-c^2 x^2-1}}{3 e \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rule 6302
Rule 12
Rule 573
Rule 152
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-2 d-3 e x^2}{3 e^2 x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-2 d-3 e x^2}{x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2 \sqrt{-c^2 x^2}}\\ &=\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{-2 d-3 e x}{x \sqrt{-1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c x \sqrt{-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{d \left (c^2 d-e\right )}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 d \left (c^2 d-e\right ) e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c x \sqrt{-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c x \sqrt{-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{(2 b c x) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1-c^2 x^2}}\right )}{3 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c x \sqrt{-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{2 b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{3 \sqrt{d} e^2 \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.284124, size = 201, normalized size = 1.19 \[ \frac{a \left (c^2 d-e\right ) \left (2 d+3 e x^2\right )+b c e x \sqrt{\frac{1}{c^2 x^2}+1} \left (d+e x^2\right )+b \left (c^2 d-e\right ) \text{csch}^{-1}(c x) \left (2 d+3 e x^2\right )}{3 e^2 \left (e-c^2 d\right ) \left (d+e x^2\right )^{3/2}}+\frac{2 b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right )}{3 \sqrt{d} e^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \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*} -\frac{1}{3} \, a{\left (\frac{3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e} + \frac{2 \, d}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e^{2}}\right )} + b \int \frac{x^{3} \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + \frac{1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 4.10332, size = 1627, normalized size = 9.63 \begin{align*} \text{result too large to display} \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 \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]