Optimal. Leaf size=410 \[ \frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}-\frac{3 \sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{24 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{(c+d x)^2+1}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.1098, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {5865, 12, 5667, 5774, 5665, 3308, 2180, 2204, 2205, 5655, 5779} \[ \frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}-\frac{3 \sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{24 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{(c+d x)^2+1}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5865
Rule 12
Rule 5667
Rule 5774
Rule 5665
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5655
Rule 5779
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^2 x^2}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac{\left (4 e^2\right ) \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}+\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{\left (8 e^2\right ) \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}+\frac{\left (12 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{1+(c+d x)^2}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{24 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{\left (16 e^2\right ) \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}+\frac{\left (24 e^2\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{3 \sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{1+(c+d x)^2}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{24 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{\left (16 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d}-\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac{\left (18 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{1+(c+d x)^2}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{24 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{\left (8 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac{\left (8 e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}-\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}-\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{1+(c+d x)^2}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{24 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{\left (16 e^2\right ) \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}+\frac{\left (16 e^2\right ) \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}+\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac{\left (18 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac{\left (18 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{5 b^4 d}\\ &=-\frac{2 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{16 e^2 \sqrt{1+(c+d x)^2}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{24 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2}}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{e^2 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}-\frac{3 e^2 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{e^2 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}+\frac{3 e^2 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}\\ \end{align*}
Mathematica [A] time = 1.44251, size = 474, normalized size = 1.16 \[ \frac{e^2 \left (e^{-\sinh ^{-1}(c+d x)} \left (-4 e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+4 a^2+2 b (4 a-b) \sinh ^{-1}(c+d x)-2 a b+4 b^2 \sinh ^{-1}(c+d x)^2+3 b^2\right )-3 \left (2 e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right ) \left (6 \sqrt{3} b \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+e^{3 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )} \left (6 a+6 b \sinh ^{-1}(c+d x)+b\right )\right )+b^2 e^{3 \sinh ^{-1}(c+d x)}\right )-3 e^{-3 \sinh ^{-1}(c+d x)} \left (2 \left (a+b \sinh ^{-1}(c+d x)\right ) \left (-6 \sqrt{3} e^{3 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+6 a+6 b \sinh ^{-1}(c+d x)-b\right )+b^2\right )+2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right ) \left (2 b \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )+e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (2 a+2 b \sinh ^{-1}(c+d x)+b\right )\right )+3 b^2 e^{\sinh ^{-1}(c+d x)}\right )}{60 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.208, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dex+ce \right ) ^{2} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{2}}{{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[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 (d e x + c e\right )}^{2}}{{\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.
[In]
[Out]