Optimal. Leaf size=505 \[ -\frac {c d e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {i c d e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+b c d e \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )-\frac {d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {2 b c d e \sqrt {c d x+d} \sqrt {e-c e x} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 b c^3 d e x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {i b^2 c d e \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 b^2 c d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} b^2 c^2 d e x \sqrt {c d x+d} \sqrt {e-c e x} \]
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Rubi [A] time = 0.81, antiderivative size = 505, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4739, 4695, 4647, 4641, 4627, 321, 216, 4683, 4625, 3717, 2190, 2279, 2391, 195} \[ -\frac {i b^2 c d e \sqrt {c d x+d} \sqrt {e-c e x} \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {c d e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {i c d e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+b c d e \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )-\frac {d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {2 b c d e \sqrt {c d x+d} \sqrt {e-c e x} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {5 b^2 c d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} b^2 c^2 d e x \sqrt {c d x+d} \sqrt {e-c e x} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 321
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4627
Rule 4641
Rule 4647
Rule 4683
Rule 4695
Rule 4739
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (3 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (3 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {1}{2} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \operatorname {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^4 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{2 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {\left (4 i b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (i b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 2.25, size = 538, normalized size = 1.07 \[ \frac {12 a^2 c d^{3/2} e^{3/2} x \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )-4 a^2 c^2 d e x^2 \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x}-8 a^2 d e \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x}-2 b d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (6 a c x+4 b \sqrt {1-c^2 x^2}+4 i b c x+b c x \sin \left (2 \sin ^{-1}(c x)\right )\right )-2 b d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (8 a \sqrt {1-c^2 x^2}+2 a c x \sin \left (2 \sin ^{-1}(c x)\right )-8 b c x \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b c x \cos \left (2 \sin ^{-1}(c x)\right )\right )+16 a b c d e x \sqrt {c d x+d} \sqrt {e-c e x} \log (c x)-2 a b c d e x \sqrt {c d x+d} \sqrt {e-c e x} \cos \left (2 \sin ^{-1}(c x)\right )-8 i b^2 c d e x \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-4 b^2 c d e x \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3+b^2 c d e x \sqrt {c d x+d} \sqrt {e-c e x} \sin \left (2 \sin ^{-1}(c x)\right )}{8 x \sqrt {1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c^{2} d e x^{2} - a^{2} d e + {\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d e x^{2} - a b d e\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{x^{2}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (3 \, \sqrt {-c^{2} d e x^{2} + d e} c^{2} d e x + \frac {3 \, c d^{2} e^{2} \arcsin \left (c x\right )}{\sqrt {d e}} + \frac {2 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}}}{x}\right )} a^{2} - \sqrt {d} \sqrt {e} \int \frac {{\left ({\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d e x^{2} - a b d e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}}{x^2} \,d x \]
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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