Optimal. Leaf size=128 \[ -\frac{b c d (3 m+7) x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2}+\frac{c^2 d x^{m+3} \left (a+b \sinh ^{-1}(c x)\right )}{m+3}+\frac{d x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{m+1}-\frac{b c d \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.129572, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {14, 5730, 12, 459, 364} \[ \frac{c^2 d x^{m+3} \left (a+b \sinh ^{-1}(c x)\right )}{m+3}+\frac{d x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{m+1}-\frac{b c d (3 m+7) x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2}-\frac{b c d \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 5730
Rule 12
Rule 459
Rule 364
Rubi steps
\begin{align*} \int x^m \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{d x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{c^2 d x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}-(b c) \int \frac{d x^{1+m} \left (\frac{1}{1+m}+\frac{c^2 x^2}{3+m}\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=\frac{d x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{c^2 d x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}-(b c d) \int \frac{x^{1+m} \left (\frac{1}{1+m}+\frac{c^2 x^2}{3+m}\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{b c d x^{2+m} \sqrt{1+c^2 x^2}}{(3+m)^2}+\frac{d x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{c^2 d x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}-\frac{(b c d (7+3 m)) \int \frac{x^{1+m}}{\sqrt{1+c^2 x^2}} \, dx}{(1+m) (3+m)^2}\\ &=-\frac{b c d x^{2+m} \sqrt{1+c^2 x^2}}{(3+m)^2}+\frac{d x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac{c^2 d x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}-\frac{b c d (7+3 m) x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-c^2 x^2\right )}{(1+m) (2+m) (3+m)^2}\\ \end{align*}
Mathematica [A] time = 0.0686725, size = 118, normalized size = 0.92 \[ \frac{d x^{m+1} \left (-b c (m+1) x \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )-2 b c x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )+(m+2) \left (c^2 m x^2+c^2 x^2+m+3\right ) \left (a+b \sinh ^{-1}(c x)\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.754, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({c}^{2}d{x}^{2}+d \right ) \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a c^{2} d x^{2} + a d +{\left (b c^{2} d x^{2} + b d\right )} \operatorname{arsinh}\left (c x\right )\right )} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d \left (\int a x^{m}\, dx + \int b x^{m} \operatorname{asinh}{\left (c x \right )}\, dx + \int a c^{2} x^{2} x^{m}\, dx + \int b c^{2} x^{2} x^{m} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \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 (c^{2} d x^{2} + d\right )}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]