Optimal. Leaf size=411 \[ \frac {\sqrt {\pi } c^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\pi } c^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} c e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{2}} c e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{3}} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3} \]
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Rubi [A] time = 0.86, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5865, 5805, 6741, 6742, 5299, 2205, 2204, 5298, 5618} \[ \frac {\sqrt {\pi } c^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\pi } c^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{2}} c e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}-\frac {\sqrt {\frac {\pi }{2}} c e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {\sqrt {\frac {\pi }{3}} e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5299
Rule 5618
Rule 5805
Rule 5865
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b \sinh ^{-1}(c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {c}{d}+\frac {x}{d}\right )^2}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right )^2}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (c^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+c \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+\cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac {(2 c) \operatorname {Subst}\left (\int \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{4} \cosh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac {c \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}-\frac {c \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac {c^2 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac {c^2 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\operatorname {Subst}\left (\int \cosh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 b d^3}-\frac {\operatorname {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 b d^3}\\ &=\frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {\operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}-\frac {\operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}-\frac {\operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}+\frac {\operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}\\ &=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}+\frac {c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}-\frac {c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 \sqrt {b} d^3}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 471, normalized size = 1.15 \[ \frac {\sqrt {\pi } \left (3 \left (4 c^2-1\right ) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-12 c^2 \sinh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+12 c^2 \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {3} \sinh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+6 \sqrt {2} c \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {3} \cosh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+6 \sqrt {2} c \sinh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+3 \sinh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\sqrt {3} \sinh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-6 \sqrt {2} c \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {3} \cosh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )\right )}{24 \sqrt {b} d^3} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {x^{2}}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {x^{2}}{\sqrt {a +b \arcsinh \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {x^{2}}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {x^2}{\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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