Optimal. Leaf size=249 \[ \frac {30 b^2 \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{d x}-\frac {5 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^{3/2}}{d x}+\frac {30 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2} d x}-\frac {30 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2} d x}+x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^{5/2} \]
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Rubi [A] time = 0.09, antiderivative size = 249, 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 = {4815, 4812} \[ \frac {30 b^2 \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{d x}-\frac {5 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^{3/2}}{d x}+\frac {30 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2} d x}-\frac {30 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2} d x}+x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 4812
Rule 4815
Rubi steps
\begin {align*} \int \left (a+b \cos ^{-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 \cos ^{-1}\left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^{5/2}-\left (15 b^2\right ) \int \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )} \, dx\\ &=-\frac {5 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^{5/2}-\frac {30 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{\left (\frac {1}{b}\right )^{5/2} d x}+\frac {30 \sqrt {\pi } C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{\left (\frac {1}{b}\right )^{5/2} d x}+\frac {30 b^2 \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )} \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{d x}\\ \end {align*}
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Mathematica [A] time = 2.75, size = 256, normalized size = 1.03 \[ -\frac {2 \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \left (\sqrt {a+b \cos ^{-1}\left (d x^2+1\right )} \left (\left (a^2-15 b^2\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right )+5 a b \cos \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right )+b \cos ^{-1}\left (d x^2+1\right ) \left (2 a \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right )+5 b \cos \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right )\right )+b^2 \cos ^{-1}\left (d x^2+1\right )^2 \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right )\right )-\frac {15 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2}}+\frac {15 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\left (\frac {1}{b}\right )^{5/2}}\right )}{d x} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^{5/2} \,d x \]
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 \[ \int \left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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