Optimal. Leaf size=496 \[ \frac {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d^3} \]
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Rubi [A]
time = 1.39, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5859, 5830,
6873, 6874, 5433, 5406, 2236, 2235, 5432, 5407, 5480, 5408} \begin {gather*} \frac {\sqrt {\pi } \sqrt {b} c^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\pi } \sqrt {b} c^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}-\frac {\sqrt {\pi } \sqrt {b} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 5406
Rule 5407
Rule 5408
Rule 5432
Rule 5433
Rule 5480
Rule 5830
Rule 5859
Rule 6873
Rule 6874
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b \sinh ^{-1}(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right )^2 \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \sqrt {a+b x} \cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right )^2 \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int x^2 \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 \text {Subst}\left (\int x^2 \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 \text {Subst}\left (\int \left (c^2 x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+c x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+x^2 \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 \text {Subst}\left (\int x^2 \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) \text {Subst}\left (\int x^2 \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 ) \text {Subst}\left (\int x^2 \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 {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac {\text {Subst}\left (\int \sinh ^3\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{3 d^3}+\frac {c \text {Subst}\left (\int \cosh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 d^3}+\frac {c^2 \text {Subst}\left (\int \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac {\text {Subst}\left (\int \left (\frac {1}{4} \sinh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )-\frac {3}{4} \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{3 d^3}+\frac {c \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 d^3}+\frac {c \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 d^3}+\frac {c^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 d^3}-\frac {c^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} 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 )}{8 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} 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 )}{8 d^3}+\frac {\text {Subst}\left (\int \sinh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{12 d^3}-\frac {\text {Subst}\left (\int \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} 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 )}{8 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} 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 )}{8 d^3}+\frac {\text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{24 d^3}-\frac {\text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{24 d^3}-\frac {\text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{8 d^3}+\frac {\text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{8 d^3}\\ &=\frac {c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{2 d^3}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} 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 )}{8 d^3}+\frac {\sqrt {b} 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 )}{48 d^3}+\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} 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 )}{8 d^3}-\frac {\sqrt {b} 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 )}{48 d^3}\\ \end {align*}
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Mathematica [A]
time = 1.17, size = 656, normalized size = 1.32 \begin {gather*} \frac {-36 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}+144 c^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}-72 c \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )+\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-36 \sqrt {b} c^2 \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} c \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )-9 \sqrt {b} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+36 \sqrt {b} c^2 \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+9 \sqrt {b} \left (-1+4 c^2\right ) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )-9 \sqrt {b} c \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+9 \sqrt {b} c \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {b} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+\sqrt {b} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+12 \sqrt {a+b \sinh ^{-1}(c+d x)} \sinh \left (3 \sinh ^{-1}(c+d x)\right )}{144 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {a +b \arcsinh \left (d x +c \right )}\, dx\]
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 \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*} \int x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx \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 x^2\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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