Optimal. Leaf size=277 \[ -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{4 b \sqrt {1-c^2 x^2}}-\frac {7 b c^3 d^2 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {4785, 4741,
4737, 30, 14, 272, 45} \begin {gather*} \frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{4 b \sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {7 b c^3 d^2 \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 30
Rule 45
Rule 272
Rule 4737
Rule 4741
Rule 4785
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {1}{3} \left (5 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1-c^2 x\right )^2}{x^2} \, dx,x,x^2\right )}{6 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (c^4+\frac {1}{x^2}-\frac {2 c^2}{x}\right ) \, dx,x,x^2\right )}{6 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {1}{x}-c^2 x\right ) \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (5 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b \sqrt {1-c^2 x^2}}-\frac {7 b c^3 d^2 \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 = 0.93, size = 243, normalized size = 0.88 \begin {gather*} \frac {1}{24} d^2 \left (\frac {4 b \sqrt {d-c^2 d x^2} \left (-2+14 c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)}{x^3}+\frac {30 b c^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2}{\sqrt {1-c^2 x^2}}-60 a c^3 \sqrt {d} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {\sqrt {d-c^2 d x^2} \left (4 a \sqrt {1-c^2 x^2} \left (-2+14 c^2 x^2+3 c^4 x^4\right )+b \left (-4 c x+3 c^3 x^3-6 c^5 x^5\right )-56 b c^3 x^3 \log (c x)\right )}{x^3 \sqrt {1-c^2 x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.32, size = 1527, normalized size = 5.51
method | result | size |
default | \(\text {Expression too large to display}\) | \(1527\) |
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 {5}{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 = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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