Optimal. Leaf size=365 \[ \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2} \]
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Rubi [A]
time = 0.61, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 15, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used =
{5859, 5829, 5773, 5818, 5774, 3388, 2211, 2236, 2235, 5779, 5780, 5556, 12, 3389, 5783}
\begin {gather*} -\frac {2 \sqrt {\pi } c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 \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 )}{3 b^{5/2} d^2}-\frac {2 \sqrt {\pi } c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {2 \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 )}{3 b^{5/2} d^2}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {2 c \sqrt {(c+d x)^2+1}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3389
Rule 5556
Rule 5773
Rule 5774
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rule 5829
Rule 5859
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{\left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {c}{d \left (a+b \sinh ^{-1}(x)\right )^{5/2}}+\frac {x}{d \left (a+b \sinh ^{-1}(x)\right )^{5/2}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}+\frac {4 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}-\frac {(2 c) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {16 \text {Subst}\left (\int \frac {x}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d^2}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d^2}-\frac {(4 c) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{3 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {16 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d^2}-\frac {(2 c) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{3 b^3 d^2}-\frac {(2 c) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{3 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {8 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d^2}-\frac {(4 c) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{3 b^3 d^2}-\frac {(4 c) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{3 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {4 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d^2}+\frac {4 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {8 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{3 b^3 d^2}+\frac {8 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{3 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 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 )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {2 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 )}{3 b^{5/2} d^2}\\ \end {align*}
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Mathematica [A]
time = 1.81, size = 375, normalized size = 1.03 \begin {gather*} \frac {\frac {\sqrt {b} c e^{-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \left (e^{a/b} \left (-2 a+b+2 a e^{2 \sinh ^{-1}(c+d x)}+b e^{2 \sinh ^{-1}(c+d x)}+2 b \left (-1+e^{2 \sinh ^{-1}(c+d x)}\right ) \sinh ^{-1}(c+d x)\right )+2 e^{\frac {2 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 ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+2 b e^{\sinh ^{-1}(c+d x)} \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+2 \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )-2 \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {\sqrt {b} \left (4 \left (a+b \sinh ^{-1}(c+d x)\right ) \cosh \left (2 \sinh ^{-1}(c+d x)\right )+b \sinh \left (2 \sinh ^{-1}(c+d x)\right )\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}}{3 b^{5/2} d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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