Optimal. Leaf size=346 \[ -\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}-\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3} \]
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Rubi [A]
time = 0.69, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5779, 5818,
5778, 3389, 2211, 2236, 2235, 5773, 5819} \begin {gather*} \frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}-\frac {3 \sqrt {3 \pi } e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}-\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}-\frac {24 x^2 \sqrt {c^2 x^2+1}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {16 \sqrt {c^2 x^2+1}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5773
Rule 5778
Rule 5779
Rule 5818
Rule 5819
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{7/2}} \, dx &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}+\frac {4 \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b c}+\frac {(6 c) \int \frac {x^3}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b}\\ &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac {12 \int \frac {x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{5 b^2}+\frac {8 \int \frac {1}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx}{15 b^2 c^2}\\ &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {24 \text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {3 \sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {16 \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{15 b^3 c}\\ &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {16 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^3}-\frac {6 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {18 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {8 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac {8 \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}-\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}-\frac {9 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {16 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{15 b^4 c^3}+\frac {16 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{15 b^4 c^3}+\frac {6 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}-\frac {6 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}-\frac {18 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {18 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{5 b^4 c^3}\\ &=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c \left (a+b \sinh ^{-1}(c x)\right )^{5/2}}-\frac {8 x}{15 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}-\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}\\ \end {align*}
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Mathematica [A]
time = 1.07, size = 417, normalized size = 1.21 \begin {gather*} \frac {3 b^2 e^{\sinh ^{-1}(c x)}+e^{-\sinh ^{-1}(c x)} \left (4 a^2-2 a b+3 b^2+2 (4 a-b) b \sinh ^{-1}(c x)+4 b^2 \sinh ^{-1}(c x)^2-4 e^{\frac {a}{b}+\sinh ^{-1}(c x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right )^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 \left (b^2 e^{3 \sinh ^{-1}(c x)}+2 e^{-\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c x)\right ) \left (e^{3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \left (6 a+b+6 b \sinh ^{-1}(c x)\right )+6 \sqrt {3} b \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )+2 e^{-\frac {a}{b}} \left (a+b \sinh ^{-1}(c x)\right ) \left (e^{\frac {a}{b}+\sinh ^{-1}(c x)} \left (2 a+b+2 b \sinh ^{-1}(c x)\right )+2 b \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )\right )-3 e^{-3 \sinh ^{-1}(c x)} \left (b^2+2 \left (a+b \sinh ^{-1}(c x)\right ) \left (6 a-b+6 b \sinh ^{-1}(c x)-6 \sqrt {3} e^{3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \left (a+b \sinh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{60 b^3 c^3 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {7}{2}}}\, dx\]
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*} \int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {7}{2}}}\, 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 \frac {x^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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