Optimal. Leaf size=191 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.17, 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 = {4785, 4781, 29,
4737, 14} \begin {gather*} \frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 4737
Rule 4781
Rule 4785
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 = 0.48, size = 211, normalized size = 1.10 \begin {gather*} \frac {b d \left (-1+4 c^2 x^2\right ) \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{3 x^3}+\frac {b c^3 d \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2}{2 \sqrt {1-c^2 x^2}}-a c^3 d^{3/2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {d \sqrt {d-c^2 d x^2} \left (b c x+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)\right )}{6 x^3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 1289, normalized size = 6.75
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d \,c^{3}}{2 \left (c^{2} x^{2}-1\right )}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {10 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} c^{6}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {32 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{5} \arcsin \left (c x \right ) c^{8}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {8 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{5} c^{8}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {8 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \,c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {52 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{3} \arcsin \left (c x \right ) c^{6}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {32 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{7}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {12 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2} \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {8 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) d \,c^{3}}{3 c^{2} x^{2}-3}+\frac {73 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \arcsin \left (c x \right ) c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}+\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d x \left (-c^{2} x^{2}+1\right ) c^{4}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, c^{3}}{2 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-1\right )}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right ) c^{2}}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \arcsin \left (c x \right )}{3 \left (24 c^{4} x^{4}-9 c^{2} x^{2}+1\right ) x^{3} \left (c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) d \,c^{3}}{3 \left (c^{2} x^{2}-1\right )}\) | \(1289\) |
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 (- 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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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