Optimal. Leaf size=277 \[ -15 b^2 x \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}+\frac {5 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \text {ArcSin}\left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \text {ArcSin}\left (1+d x^2\right )\right )^{5/2}-\frac {15 \sqrt {\pi } x S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )\right )}+\frac {15 \sqrt {\pi } x \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4898, 4895}
\begin {gather*} -15 b^2 x \sqrt {a+b \text {ArcSin}\left (d x^2+1\right )}+\frac {5 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \text {ArcSin}\left (d x^2+1\right )\right )^{3/2}}{d x}+\frac {15 \sqrt {\pi } x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )\right )}-\frac {15 \sqrt {\pi } x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )\right )}+x \left (a+b \text {ArcSin}\left (d x^2+1\right )\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 4895
Rule 4898
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{5/2} \, dx &=\frac {5 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{5/2}-\left (15 b^2\right ) \int \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )} \, dx\\ &=-15 b^2 x \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )}+\frac {5 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{5/2}-\frac {15 \sqrt {\pi } x S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )\right )}+\frac {15 \sqrt {\pi } x C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 269, normalized size = 0.97 \begin {gather*} \frac {5 b \sqrt {-d x^2 \left (2+d x^2\right )} \left (a+b \text {ArcSin}\left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \text {ArcSin}\left (1+d x^2\right )\right )^{5/2}-\frac {15 x \left (\sqrt {\pi } S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )-\sqrt {\pi } \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )+\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )\right )\right )}{\left (\frac {1}{b}\right )^{5/2} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (a +b \arcsin \left (d \,x^{2}+1\right )\right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {asin}{\left (d x^{2} + 1 \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {\left (a+b\,\mathrm {asin}\left (d\,x^2+1\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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