Optimal. Leaf size=161 \[ -\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {a^2 c x^2+c}}+\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4930, 4898, 4905, 4904, 3304, 3352} \[ -\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {a^2 c x^2+c}}+\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3352
Rule 4898
Rule 4904
Rule 4905
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {5 \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{2 a}\\ &=\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {c+a^2 c x^2}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {15 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{8 a}\\ &=\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {c+a^2 c x^2}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {\left (15 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{8 a c \sqrt {c+a^2 c x^2}}\\ &=\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {c+a^2 c x^2}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {\left (15 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c \sqrt {c+a^2 c x^2}}\\ &=\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {c+a^2 c x^2}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {\left (15 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c \sqrt {c+a^2 c x^2}}\\ &=\frac {15 \sqrt {\tan ^{-1}(a x)}}{4 a^2 c \sqrt {c+a^2 c x^2}}+\frac {5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.19, size = 139, normalized size = 0.86 \[ \frac {15 i \sqrt {a^2 x^2+1} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \tan ^{-1}(a x)\right )-15 i \sqrt {a^2 x^2+1} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \tan ^{-1}(a x)\right )+4 \tan ^{-1}(a x) \left (-4 \tan ^{-1}(a x)^2+10 a x \tan ^{-1}(a x)+15\right )}{16 a^2 c \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 3.01, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________