Optimal. Leaf size=82 \[ \frac {6 b \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {(c+d x)^2+1} (e (c+d x))^{5/2}},x\right )}{5 e}-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{(e x)^{5/2} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d e}\\ \end {align*}
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Mathematica [A] time = 73.71, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arsinh}\left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b^{3} \sqrt {e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3}}{5 \, {\left (d^{3} e^{4} x^{2} + 2 \, c d^{2} e^{4} x + c^{2} d e^{4}\right )} \sqrt {d x + c}} - \frac {2 \, a^{3}}{5 \, {\left (d e x + c e\right )}^{\frac {5}{2}} d e} + \int \frac {3 \, {\left ({\left (2 \, {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} b^{3} + {\left (5 \, a b^{2} d^{3} \sqrt {e} + 2 \, b^{3} d^{3} \sqrt {e}\right )} x^{3} + 5 \, {\left (a c^{3} \sqrt {e} + a c \sqrt {e}\right )} b^{2} + 3 \, {\left (5 \, a b^{2} c d^{2} \sqrt {e} + 2 \, b^{3} c d^{2} \sqrt {e}\right )} x^{2} + {\left (2 \, {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} b^{3} + 5 \, {\left (3 \, a c^{2} d \sqrt {e} + a d \sqrt {e}\right )} b^{2}\right )} x + {\left (2 \, b^{3} c^{2} \sqrt {e} + 5 \, {\left (a c^{2} \sqrt {e} + a \sqrt {e}\right )} b^{2} + {\left (5 \, a b^{2} d^{2} \sqrt {e} + 2 \, b^{3} d^{2} \sqrt {e}\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c d \sqrt {e} + 2 \, b^{3} c d \sqrt {e}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 5 \, {\left (a^{2} b d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, a^{2} c^{2} d \sqrt {e} + a^{2} d \sqrt {e}\right )} b x + {\left (a^{2} c^{3} \sqrt {e} + a^{2} c \sqrt {e}\right )} b + {\left (a^{2} b d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b c d \sqrt {e} x + {\left (a^{2} c^{2} \sqrt {e} + a^{2} \sqrt {e}\right )} b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )}}{5 \, {\left ({\left (d^{5} e^{4} x^{5} + 5 \, c d^{4} e^{4} x^{4} + c^{5} e^{4} + c^{3} e^{4} + {\left (10 \, c^{2} d^{3} e^{4} + d^{3} e^{4}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{4} + 3 \, c d^{2} e^{4}\right )} x^{2} + {\left (5 \, c^{4} d e^{4} + 3 \, c^{2} d e^{4}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + c^{6} e^{4} + c^{4} e^{4} + {\left (15 \, c^{2} d^{4} e^{4} + d^{4} e^{4}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{4} + c d^{3} e^{4}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{4} + 2 \, c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{5} d e^{4} + 2 \, c^{3} d e^{4}\right )} x\right )} \sqrt {d x + c}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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