Optimal. Leaf size=349 \[ -\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac {b^2 c^3 d \text {PolyLog}\left (2,-\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {PolyLog}\left (2,-\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.42, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5828, 5843,
3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {b c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {e e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 5828
Rule 5843
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{(d+e x)^2 \sqrt {1+c^2 x^2}} \, dx}{e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {a+b x}{(c d+e \sinh (x))^2} \, dx,x,\sinh ^{-1}(c x)\right )}{e}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\cosh (x)}{c d+e \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d^2+e^2}+\frac {\left (b c^3 d\right ) \text {Subst}\left (\int \frac {a+b x}{c d+e \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e \left (c^2 d^2+e^2\right )}+\frac {\left (2 b c^3 d\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{-e+2 c d e^x+e e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac {\left (2 b c^3 d\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d-2 \sqrt {c^2 d^2+e^2}+2 e e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^{3/2}}-\frac {\left (2 b c^3 d\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d+2 \sqrt {c^2 d^2+e^2}+2 e e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}-\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d-2 \sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d+2 \sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}-\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d-2 \sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d+2 \sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 270, normalized size = 0.77 \begin {gather*} \frac {-\frac {2 b c e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{(d+e x)^2}+\frac {2 b^2 c^2 \log (d+e x)}{c^2 d^2+e^2}+\frac {2 b c^3 d \left (\left (a+b \sinh ^{-1}(c x)\right ) \left (\log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-\log \left (1+\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )+b \text {PolyLog}\left (2,\frac {e e^{\sinh ^{-1}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-b \text {PolyLog}\left (2,-\frac {e e^{\sinh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )}{\left (c^2 d^2+e^2\right )^{3/2}}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1016\) vs.
\(2(365)=730\).
time = 12.87, size = 1017, normalized size = 2.91 Too large to display
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, 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 {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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