Optimal. Leaf size=391 \[ -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {6} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {6} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5790, 5819,
5556, 3389, 2211, 2235, 2236} \begin {gather*} -\frac {3 \sqrt {\pi } c^2 \sqrt {a^2 c x^2+c} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {a^2 x^2+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}-\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {6} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\pi } c^2 \sqrt {a^2 c x^2+c} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {a^2 x^2+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {6} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \left (a^2 c x^2+c\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5790
Rule 5819
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (12 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {x \left (1+a^2 x^2\right )^2}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{\sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (12 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^5(x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (12 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}+\frac {\sinh (6 x)}{32 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (6 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a \sqrt {1+a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{2 a \sqrt {1+a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{5/2}}{a \sqrt {\sinh ^{-1}(a x)}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}+\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 399, normalized size = 1.02 \begin {gather*} \frac {c^2 e^{-6 \sinh ^{-1}(a x)} \sqrt {c+a^2 c x^2} \left (-1-6 e^{2 \sinh ^{-1}(a x)}+e^{4 \sinh ^{-1}(a x)}-52 e^{6 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}-6 e^{10 \sinh ^{-1}(a x)}-e^{12 \sinh ^{-1}(a x)}-64 a^2 e^{6 \sinh ^{-1}(a x)} x^2-16 e^{6 \sinh ^{-1}(a x)} \sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+16 e^{6 \sinh ^{-1}(a x)} \sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+\sqrt {6} e^{6 \sinh ^{-1}(a x)} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-6 \sinh ^{-1}(a x)\right )+12 e^{6 \sinh ^{-1}(a x)} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )-\sqrt {2} e^{6 \sinh ^{-1}(a x)} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )-\sqrt {2} e^{6 \sinh ^{-1}(a x)} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )+12 e^{6 \sinh ^{-1}(a x)} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )+\sqrt {6} e^{6 \sinh ^{-1}(a x)} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},6 \sinh ^{-1}(a x)\right )\right )}{32 a \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\arcsinh \left (a x \right )^{\frac {3}{2}}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\left (c\,a^2\,x^2+c\right )}^{5/2}}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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