Optimal. Leaf size=191 \[ \frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.23, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4695, 4693, 29, 4641, 14} \[ \frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt {1-c^2 x^2}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \log (x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 4641
Rule 4693
Rule 4695
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\left (c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right ) \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 211, normalized size = 1.10 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (2 a \left (1-4 c^2 x^2\right ) \sqrt {1-c^2 x^2}+8 b c^3 x^3 \log (c x)+b c x\right )}{6 x^3 \sqrt {1-c^2 x^2}}-a c^3 d^{3/2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+\frac {b d \left (4 c^2 x^2-1\right ) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 x^3}+\frac {b c^3 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)^2}{2 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a c^{2} d x^{2} - a d + {\left (b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [C] time = 0.41, size = 1289, normalized size = 6.75 \[ \text {result too large to display} \]
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 \[ -b \sqrt {d} \int \frac {{\left (c^{2} d x^{2} - d\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x^{4}}\,{d x} + \frac {1}{3} \, {\left (3 \, \sqrt {-c^{2} d x^{2} + d} c^{4} d x + 3 \, c^{3} d^{\frac {3}{2}} \arcsin \left (c x\right ) + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{x} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{3}}\right )} a \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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