Optimal. Leaf size=247 \[ \frac{3 \sqrt{\pi } b^{3/2} x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{\sqrt{\pi } \sqrt{b}}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )}+\frac{3 \sqrt{\pi } b^{3/2} x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{\sqrt{b} \sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )}+\frac{3 b \sqrt{-d^2 x^4-2 d x^2} \sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{d x}+x \left (a+b \sin ^{-1}\left (d x^2+1\right )\right )^{3/2} \]
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Rubi [A] time = 0.0757509, antiderivative size = 247, normalized size of antiderivative = 1., 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 = {4814, 4819} \[ \frac{3 \sqrt{\pi } b^{3/2} x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{\sqrt{\pi } \sqrt{b}}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )}+\frac{3 \sqrt{\pi } b^{3/2} x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{\sqrt{b} \sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )}+\frac{3 b \sqrt{-d^2 x^4-2 d x^2} \sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{d x}+x \left (a+b \sin ^{-1}\left (d x^2+1\right )\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 4814
Rule 4819
Rubi steps
\begin{align*} \int \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{3/2} \, dx &=\frac{3 b \sqrt{-2 d x^2-d^2 x^4} \sqrt{a+b \sin ^{-1}\left (1+d x^2\right )}}{d x}+x \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{3/2}-\left (3 b^2\right ) \int \frac{1}{\sqrt{a+b \sin ^{-1}\left (1+d x^2\right )}} \, dx\\ &=\frac{3 b \sqrt{-2 d x^2-d^2 x^4} \sqrt{a+b \sin ^{-1}\left (1+d x^2\right )}}{d x}+x \left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{3/2}+\frac{3 b^{3/2} \sqrt{\pi } x C\left (\frac{\sqrt{a+b \sin ^{-1}\left (1+d x^2\right )}}{\sqrt{b} \sqrt{\pi }}\right ) \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right )}{\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 )}+\frac{3 b^{3/2} \sqrt{\pi } x S\left (\frac{\sqrt{a+b \sin ^{-1}\left (1+d x^2\right )}}{\sqrt{b} \sqrt{\pi }}\right ) \left (\cos \left (\frac{a}{2 b}\right )+\sin \left (\frac{a}{2 b}\right )\right )}{\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 )}\\ \end{align*}
Mathematica [A] time = 0.360626, size = 249, normalized size = 1.01 \[ \frac{3 \sqrt{\pi } b^{3/2} x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{\sqrt{\pi } \sqrt{b}}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )}+\frac{3 \sqrt{\pi } b^{3/2} x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )}}{\sqrt{b} \sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (d x^2+1\right )\right )}+\frac{\sqrt{a+b \sin ^{-1}\left (d x^2+1\right )} \left (a d x^2+3 b \sqrt{-d x^2 \left (d x^2+2\right )}+b d x^2 \sin ^{-1}\left (d x^2+1\right )\right )}{d x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arcsin \left ( d{x}^{2}+1 \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x^{2} + 1\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{asin}{\left (d x^{2} + 1 \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x^{2} + 1\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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