Optimal. Leaf size=358 \[ -\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {Erf}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2} \]
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Rubi [A]
time = 0.84, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps
used = 32, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5814, 5791,
3393, 3388, 2211, 2236, 2235, 5819, 5556} \begin {gather*} \frac {\sqrt {\pi } d^2 e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 \sqrt {\frac {\pi }{2}} d^2 e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{\frac {6 a}{b}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {\sqrt {\pi } d^2 e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 \sqrt {\frac {\pi }{2}} d^2 e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {\sqrt {\frac {3 \pi }{2}} d^2 e^{-\frac {6 a}{b}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}-\frac {2 d^2 x \left (c^2 x^2+1\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5556
Rule 5791
Rule 5814
Rule 5819
Rubi steps
\begin {align*} \int \frac {x \left (d+c^2 d x^2\right )^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (2 d^2\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac {\left (12 c d^2\right ) \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {\left (12 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {a+b x}}+\frac {\cosh (2 x)}{2 \sqrt {a+b x}}+\frac {\cosh (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {\left (12 d^2\right ) \text {Subst}\left (\int \left (-\frac {1}{16 \sqrt {a+b x}}-\frac {\cosh (2 x)}{32 \sqrt {a+b x}}+\frac {\cosh (4 x)}{16 \sqrt {a+b x}}+\frac {\cosh (6 x)}{32 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d^2 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cosh (6 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d^2 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}\\ &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d^2 \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {6 a}{b}-\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {6 a}{b}+\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {d^2 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d^2 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 d^2 x \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {5 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}+\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^2}\\ \end {align*}
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Mathematica [A]
time = 3.32, size = 475, normalized size = 1.33 \begin {gather*} \frac {d^2 e^{-\frac {6 a}{b}} \left (16 e^{\frac {8 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+16 e^{\frac {4 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )-\frac {\sqrt {b} \left (64 c e^{\frac {6 a}{b}} x \sqrt {1+c^2 x^2}+128 c^3 e^{\frac {6 a}{b}} x^3 \sqrt {1+c^2 x^2}+64 c^5 e^{\frac {6 a}{b}} x^5 \sqrt {1+c^2 x^2}-\sqrt {6} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-8 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+11 \sqrt {2} e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-11 \sqrt {2} e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+8 e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt {6} e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{\sqrt {a+b \sinh ^{-1}(c x)}}\right )}{32 b^{3/2} c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 (d\,c^2\,x^2+d\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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