Optimal. Leaf size=109 \[ \frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3}-\frac {8 b c \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4627, 4711} \[ \frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3}-\frac {8 b c \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4711
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(d x)^{5/2}} \, dx &=-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}}+\frac {(4 b c) \int \frac {a+b \sin ^{-1}(c x)}{(d x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 d}\\ &=-\frac {2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}}-\frac {8 b c \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right )}{3 d^2 \sqrt {d x}}+\frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 87, normalized size = 0.80 \[ \frac {x \left (16 b^2 c^2 x^2 \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )-2 \left (a+b \sin ^{-1}(c x)\right ) \left (a+4 b c x \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right )+b \sin ^{-1}(c x)\right )\right )}{3 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {d x}}{d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (d x \right )^{\frac {5}{2}}}\, dx \]
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 \[ -\frac {{\left (3 \, a^{2} c^{2} \sqrt {d} {\left (\frac {2 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{\sqrt {c} d^{3}} - \frac {\log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{\sqrt {c} d^{3}}\right )} - 12 \, a b c^{2} \sqrt {d} \int \frac {x^{\frac {5}{2}} \arctan \left (\frac {c x}{\sqrt {c x + 1} \sqrt {-c x + 1}}\right )}{c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x} + 8 \, b^{2} c \sqrt {d} \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{\frac {3}{2}} \arctan \left (\frac {c x}{\sqrt {c x + 1} \sqrt {-c x + 1}}\right )}{c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x} - a^{2} \sqrt {d} {\left (\frac {6 \, c^{\frac {3}{2}} \arctan \left (\sqrt {c} \sqrt {x}\right )}{d^{3}} - \frac {3 \, c^{\frac {3}{2}} \log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{d^{3}} - \frac {4}{d^{3} x^{\frac {3}{2}}}\right )} + 12 \, a b \sqrt {d} \int \frac {\sqrt {x} \arctan \left (\frac {c x}{\sqrt {c x + 1} \sqrt {-c x + 1}}\right )}{c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x}\right )} d^{\frac {5}{2}} x^{\frac {3}{2}} + 4 \, b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}}{6 \, d^{\frac {5}{2}} x^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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