Optimal. Leaf size=136 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right )}{3 c^2 \sqrt{d}}-\frac{a \sqrt{1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.153341, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {192, 191, 4666, 12, 571, 78, 63, 217, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right )}{3 c^2 \sqrt{d}}-\frac{a \sqrt{1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 4666
Rule 12
Rule 571
Rule 78
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+a \int \frac{x \left (3 c+2 d x^2\right )}{3 c^2 \sqrt{1-a^2 x^2} \left (c+d x^2\right )^{3/2}} \, dx\\ &=\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{a \int \frac{x \left (3 c+2 d x^2\right )}{\sqrt{1-a^2 x^2} \left (c+d x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{a \operatorname{Subst}\left (\int \frac{3 c+2 d x}{\sqrt{1-a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c^2}\\ &=-\frac{a \sqrt{1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-a^2 x} \sqrt{c+d x}} \, dx,x,x^2\right )}{3 c^2}\\ &=-\frac{a \sqrt{1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d}{a^2}-\frac{d x^2}{a^2}}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{3 a c^2}\\ &=-\frac{a \sqrt{1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{d x^2}{a^2}} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c+d x^2}}\right )}{3 a c^2}\\ &=-\frac{a \sqrt{1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right )}{3 c^2 \sqrt{d}}\\ \end{align*}
Mathematica [C] time = 0.187612, size = 120, normalized size = 0.88 \[ \frac{a x^2 \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;a^2 x^2,-\frac{d x^2}{c}\right )-\frac{a c \sqrt{1-a^2 x^2} \left (c+d x^2\right )}{a^2 c+d}+\cos ^{-1}(a x) \left (3 c x+2 d x^3\right )}{3 c^2 \left (c+d x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.183, size = 0, normalized size = 0. \begin{align*} \int{\arccos \left ( ax \right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.24082, size = 1192, normalized size = 8.76 \begin{align*} \left [-\frac{{\left (a^{2} c^{3} +{\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt{-d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \,{\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{d x^{2} + c} \sqrt{-d} + d^{2}\right ) - 2 \, \sqrt{d x^{2} + c}{\left ({\left (2 \,{\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \arccos \left (a x\right ) -{\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt{-a^{2} x^{2} + 1}\right )}}{6 \,{\left (a^{2} c^{5} d + c^{4} d^{2} +{\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, -\frac{{\left (a^{2} c^{3} +{\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt{d} \arctan \left (\frac{{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{d x^{2} + c} \sqrt{d}}{2 \,{\left (a^{3} d^{2} x^{4} - a c d +{\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right ) - \sqrt{d x^{2} + c}{\left ({\left (2 \,{\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \arccos \left (a x\right ) -{\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt{-a^{2} x^{2} + 1}\right )}}{3 \,{\left (a^{2} c^{5} d + c^{4} d^{2} +{\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \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 \frac{\operatorname{acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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