Optimal. Leaf size=126 \[ -\frac {16 b^2 (e (c+d x))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};(c+d x)^2\right )}{3 d e^3}+\frac {8 b \sqrt {e (c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^2}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}} \]
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Rubi [A] time = 0.20, antiderivative size = 126, 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 = {4805, 4627, 4711} \[ -\frac {16 b^2 (e (c+d x))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};(c+d x)^2\right )}{3 d e^3}+\frac {8 b \sqrt {e (c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^2}-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4711
Rule 4805
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{(c e+d e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e \sqrt {e (c+d x)}}+\frac {8 b \sqrt {e (c+d x)} \left (a+b \sin ^{-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 = 0.09, size = 104, normalized size = 0.83 \[ -\frac {2 \left (8 b^2 (c+d x)^2 \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};(c+d x)^2\right )+3 \left (a+b \sin ^{-1}(c+d x)\right ) \left (a-4 b (c+d x) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )+b \sin ^{-1}(c+d x)\right )\right )}{3 d e \sqrt {e (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2}\right )} \sqrt {d e x + c e}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{2}}{\left (d e x +c e \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{2}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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