Optimal. Leaf size=346 \[ -\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}+\frac {5 d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c} \]
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Rubi [A]
time = 0.45, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5790, 5819,
5556, 3389, 2211, 2236, 2235} \begin {gather*} -\frac {5 \sqrt {\pi } d^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 \sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {\sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}+\frac {5 \sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {\sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {2 d^2 \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 3389
Rule 5556
Rule 5790
Rule 5819
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (10 c d^2\right ) \int \frac {x \left (1+c^2 x^2\right )^{3/2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (10 d^2\right ) \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (10 d^2\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 \sqrt {a+b x}}+\frac {3 \sinh (3 x)}{16 \sqrt {a+b x}}+\frac {\sinh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c}\\ &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c}-\frac {\left (15 d^2\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c}\\ &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c}+\frac {\left (5 d^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c}-\frac {\left (15 d^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c}\\ &=-\frac {2 d^2 \left (1+c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}-\frac {5 d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}-\frac {d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c}+\frac {5 d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}+\frac {d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c}\\ \end {align*}
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Mathematica [A]
time = 1.16, size = 440, normalized size = 1.27 \begin {gather*} \frac {d^2 e^{-5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \left (-e^{\frac {5 a}{b}}-5 e^{\frac {5 a}{b}+2 \sinh ^{-1}(c x)}-10 e^{\frac {5 a}{b}+4 \sinh ^{-1}(c x)}-10 e^{\frac {5 a}{b}+6 \sinh ^{-1}(c x)}-5 e^{\frac {5 a}{b}+8 \sinh ^{-1}(c x)}-e^{\frac {5 a}{b}+10 \sinh ^{-1}(c x)}+10 e^{\frac {6 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )+\sqrt {5} e^{5 \sinh ^{-1}(c x)} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+5 \sqrt {3} e^{\frac {2 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+10 e^{\frac {4 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )+5 \sqrt {3} e^{\frac {8 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt {5} e^{5 \left (\frac {2 a}{b}+\sinh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{16 b c \sqrt {a+b \sinh ^{-1}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx\]
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 {2 c^{2} x^{2}}{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^{4}}{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 {1}{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]
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 (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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