Optimal. Leaf size=335 \[ \frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}-\frac {\sqrt {3 \pi } d e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {\sqrt {5 \pi } d e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {\sqrt {5 \pi } d e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {2 d x^2 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}} \]
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Rubi [A] time = 1.38, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5777, 5779, 5448, 3308, 2180, 2204, 2205} \[ \frac {\sqrt {\pi } d e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}-\frac {\sqrt {3 \pi } d e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {\sqrt {5 \pi } d e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {\sqrt {5 \pi } d e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {2 d x^2 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5777
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^2 \left (d+c^2 d x^2\right )}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(4 d) \int \frac {x \sqrt {1+c^2 x^2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac {(10 c d) \int \frac {x^3 \sqrt {1+c^2 x^2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(4 d) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {(10 d) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^3(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(4 d) \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {\sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {(10 d) \operatorname {Subst}\left (\int \left (-\frac {\sinh (x)}{8 \sqrt {a+b x}}-\frac {\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^3}\\ &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {d \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {d \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {d \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac {d \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {d \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {d \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac {(5 d) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {(5 d) \operatorname {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^3}+\frac {(5 d) \operatorname {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^3}-\frac {(5 d) \operatorname {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^3}+\frac {(5 d) \operatorname {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^3}-\frac {d \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}-\frac {d \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}+\frac {d \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}+\frac {d \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}+\frac {(5 d) \operatorname {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^3}-\frac {(5 d) \operatorname {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^3}\\ &=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}-\frac {d 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^3}-\frac {d 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^3}-\frac {d 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^3}+\frac {d 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^3}+\frac {d 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^3}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 436, normalized size = 1.30 \[ \frac {d e^{-5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \left (-e^{\frac {5 a}{b}+2 \sinh ^{-1}(c x)}+2 e^{\frac {5 a}{b}+4 \sinh ^{-1}(c x)}+2 e^{\frac {5 a}{b}+6 \sinh ^{-1}(c x)}-e^{\frac {5 a}{b}+8 \sinh ^{-1}(c x)}-e^{\frac {5 a}{b}+10 \sinh ^{-1}(c x)}-2 e^{\frac {6 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )+\sqrt {5} e^{5 \sinh ^{-1}(c x)} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt {3} e^{\frac {2 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-2 e^{\frac {4 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )+\sqrt {3} e^{\frac {8 a}{b}+5 \sinh ^{-1}(c x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt {5} e^{5 \left (\frac {2 a}{b}+\sinh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-e^{\frac {5 a}{b}}\right )}{16 b c^3 \sqrt {a+b \sinh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (c^{2} d x^{2} + d\right )} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (c^{2} d \,x^{2}+d \right )}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (c^{2} d x^{2} + d\right )} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,\left (d\,c^2\,x^2+d\right )}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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 \[ d \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{2} x^{4}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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