Optimal. Leaf size=349 \[ -\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {d e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {e e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \]
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Rubi [A]
time = 0.44, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5793, 5773,
5819, 3389, 2211, 2236, 2235, 5778} \begin {gather*} \frac {\sqrt {\pi } e e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {3 \pi } e e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {\pi } e e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {3 \pi } e e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {\sqrt {\pi } d e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 d \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {c^2 x^2+1}}{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 5773
Rule 5778
Rule 5793
Rule 5819
Rubi steps
\begin {align*} \int \frac {d+e x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=\int \left (\frac {d}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}+\frac {e x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}\right ) \, dx\\ &=d \int \frac {1}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx+e \int \frac {x^2}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx\\ &=-\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 c d) \int \frac {x}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}+\frac {(2 e) \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 )}{b c^3}\\ &=-\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 d) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac {e \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}\\ &=-\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {d \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}+\frac {d \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}+\frac {e \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {e \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {(3 e) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {(2 d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c}+\frac {(2 d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c}+\frac {e \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 b^2 c^3}-\frac {e \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 b^2 c^3}-\frac {(3 e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{2 b^2 c^3}\\ &=-\frac {2 d \sqrt {1+c^2 x^2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {2 e x^2 \sqrt {1+c^2 x^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 )}{b^{3/2} c}+\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}-\frac {e e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 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 )}{b^{3/2} c}-\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}\\ \end {align*}
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Mathematica [A]
time = 0.94, size = 303, normalized size = 0.87 \begin {gather*} \frac {e^{-3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \left (\left (4 c^2 d-e\right ) e^{\frac {4 a}{b}+3 \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 {3} e e^{3 \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 )+\left (4 c^2 d-e\right ) e^{\frac {2 a}{b}+3 \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 )+e^{\frac {3 a}{b}} \left (-\left (\left (1+e^{2 \sinh ^{-1}(c x)}\right ) \left (4 c^2 d e^{2 \sinh ^{-1}(c x)}+e \left (-1+e^{2 \sinh ^{-1}(c x)}\right )^2\right )\right )+\sqrt {3} e e^{3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )} \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 )\right )\right )}{4 b c^3 \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 \[\int \frac {e \,x^{2}+d}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx\]
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*} \int \frac {d + e x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{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 {e\,x^2+d}{{\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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