Optimal. Leaf size=445 \[ \frac{4 \sqrt{\pi } c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}-\frac{4 \sqrt{\pi } c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 \sqrt{(c+d x)^2+1} (c+d x)}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{(c+d x)^2+1}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{2 \sqrt{(c+d x)^2+1} (c+d x)}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac{2 c \sqrt{(c+d x)^2+1}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 1.04675, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.812, Rules used = {5865, 5803, 5655, 5774, 5779, 3308, 2180, 2204, 2205, 5667, 5665, 3307, 5675} \[ \frac{4 \sqrt{\pi } c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}-\frac{4 \sqrt{\pi } c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 \sqrt{(c+d x)^2+1} (c+d x)}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{(c+d x)^2+1}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{2 \sqrt{(c+d x)^2+1} (c+d x)}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac{2 c \sqrt{(c+d x)^2+1}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5803
Rule 5655
Rule 5774
Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5667
Rule 5665
Rule 3307
Rule 5675
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-\frac{c}{d}+\frac{x}{d}}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d \left (a+b \sinh ^{-1}(x)\right )^{7/2}}+\frac{x}{d \left (a+b \sinh ^{-1}(x)\right )^{7/2}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}+\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 \operatorname{Subst}\left (\int \frac{x}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d^2}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{32 (c+d x) \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{32 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac{(8 c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{32 (c+d x) \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac{(8 c) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{32 (c+d x) \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{32 \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}+\frac{32 \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}+\frac{(4 c) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{32 (c+d x) \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{8 e^{\frac{2 a}{b}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 e^{-\frac{2 a}{b}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{(8 c) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}-\frac{(8 c) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}\\ &=\frac{2 c \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{2 (c+d x) \sqrt{1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{8 c \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{32 (c+d x) \sqrt{1+(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{4 c e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 e^{\frac{2 a}{b}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}-\frac{4 c e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}+\frac{8 e^{-\frac{2 a}{b}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}\\ \end{align*}
Mathematica [A] time = 2.50408, size = 429, normalized size = 0.96 \[ \frac{\frac{\sqrt{b} \left (-\sinh \left (2 \sinh ^{-1}(c+d x)\right ) \left (16 \left (a+b \sinh ^{-1}(c+d x)\right )^2+3 b^2\right )-4 b \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+8 \sqrt{2 \pi } \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{2 \pi } \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d^2}-\frac{c \left (8 e^{a/b} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )-4 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right ) \left (2 b \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )+e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (2 a+2 b \sinh ^{-1}(c+d x)+b\right )\right )-2 e^{-\sinh ^{-1}(c+d x)} \left (4 a^2+2 a b \left (4 \sinh ^{-1}(c+d x)-1\right )+b^2 \left (4 \sinh ^{-1}(c+d x)^2-2 \sinh ^{-1}(c+d x)+3\right )\right )-6 b^2 e^{\sinh ^{-1}(c+d x)}\right )}{30 b^3 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.156, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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