Integrand size = 14, antiderivative size = 216 \[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\frac {15 b^2 \sqrt {a+b \arcsin (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {15 b^{5/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}-\frac {15 b^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 c^2} \]
[Out]
Time = 0.41 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4725, 4795, 4737, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int x (a+b \arcsin (c x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}-\frac {15 \sqrt {\pi } b^{5/2} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^2 \sqrt {a+b \arcsin (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2} \]
[In]
[Out]
Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4725
Rule 4737
Rule 4795
Rule 4809
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {1}{4} (5 b c) \int \frac {x^2 (a+b \arcsin (c x))^{3/2}}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {1}{16} \left (15 b^2\right ) \int x \sqrt {a+b \arcsin (c x)} \, dx-\frac {(5 b) \int \frac {(a+b \arcsin (c x))^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{8 c} \\ & = -\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}+\frac {1}{64} \left (15 b^3 c\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, dx \\ & = -\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}+\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{64 c^2} \\ & = -\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}+\frac {\left (15 b^2\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c x)\right )}{64 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \arcsin (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{128 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \arcsin (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {\left (15 b^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{128 c^2}-\frac {\left (15 b^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{128 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \arcsin (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {\left (15 b^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{64 c^2}-\frac {\left (15 b^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{64 c^2} \\ & = \frac {15 b^2 \sqrt {a+b \arcsin (c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \arcsin (c x)}+\frac {5 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{8 c}-\frac {(a+b \arcsin (c x))^{5/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{5/2}-\frac {15 b^{5/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}-\frac {15 b^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.60 \[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\frac {i b^3 e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )-e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {7}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{32 \sqrt {2} c^2 \sqrt {a+b \arcsin (c x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(170)=340\).
Time = 0.08 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.89
method | result | size |
default | \(-\frac {15 \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, b^{3}-15 \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, b^{3}+32 \arcsin \left (c x \right )^{3} \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (c x \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \arcsin \left (c x \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -30 \arcsin \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+80 \arcsin \left (c x \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+32 \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-30 \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b}{128 c^{2} \sqrt {a +b \arcsin \left (c x \right )}}\) | \(408\) |
[In]
[Out]
Exception generated. \[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\int x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
\[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\int { {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}} x \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 1307, normalized size of antiderivative = 6.05 \[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int x (a+b \arcsin (c x))^{5/2} \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2} \,d x \]
[In]
[Out]