Optimal. Leaf size=134 \[ \frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d e}-\frac {8 b (e (c+d x))^{11/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};-(c+d x)^2\right )}{99 d e^2}+\frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};-(c+d x)^2\right )}{1287 d e^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5859, 5776,
5817} \begin {gather*} \frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};-(c+d x)^2\right )}{1287 d e^3}-\frac {8 b (e (c+d x))^{11/2} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{99 d e^2}+\frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 5776
Rule 5817
Rule 5859
Rubi steps
\begin {align*} \int (c e+d e x)^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int (e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {(e x)^{9/2} \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d e}\\ &=\frac {2 (e (c+d x))^{9/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d e}-\frac {8 b (e (c+d x))^{11/2} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};-(c+d x)^2\right )}{99 d e^2}+\frac {16 b^2 (e (c+d x))^{13/2} \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};-(c+d x)^2\right )}{1287 d e^3}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 110, normalized size = 0.82 \begin {gather*} \frac {2 (e (c+d x))^{9/2} \left (143 \left (a+b \sinh ^{-1}(c+d x)\right )^2-52 b (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};-(c+d x)^2\right )+8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {13}{4},\frac {13}{4};\frac {15}{4},\frac {17}{4};-(c+d x)^2\right )\right )}{1287 d e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {7}{2}} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}\, dx\]
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]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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