Optimal. Leaf size=130 \[ -\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}}+\frac {8 b \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-(c+d x)^2\right )}{d e^2}-\frac {16 b^2 (e (c+d x))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-(c+d x)^2\right )}{3 d e^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 130, 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))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-(c+d x)^2\right )}{3 d e^3}+\frac {8 b \sqrt {e (c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-(c+d x)^2\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2}-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5776
Rule 5817
Rule 5859
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{(c e+d e x)^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}}+\frac {(4 b) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {e x} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}}+\frac {8 b \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-(c+d x)^2\right )}{d e^2}-\frac {16 b^2 (e (c+d x))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-(c+d x)^2\right )}{3 d e^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 109, normalized size = 0.84 \begin {gather*} \frac {2 \left (-3 \left (a+b \sinh ^{-1}(c+d x)\right )^2-4 b (c+d x) \left (-3 \left (a+b \sinh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-(c+d x)^2\right )+2 b (c+d x) \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};-(c+d x)^2\right )\right )\right )}{3 d e \sqrt {e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{2}}{\left (d e x +c e \right )^{\frac {3}{2}}}\, dx\]
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]
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 + d x \right )}\right )^{2}}{\left (e \left (c + d 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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