Integrand size = 23, antiderivative size = 256 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^{5/2} e \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 d} \]
[Out]
Time = 0.45 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4889, 12, 4725, 4795, 4737, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 \sqrt {\pi } b^{5/2} e \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d} \]
[In]
[Out]
Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4725
Rule 4737
Rule 4795
Rule 4809
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e x (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {x^2 (a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = \frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int x \sqrt {a+b \arcsin (x)} \, dx,x,c+d x\right )}{16 d} \\ & = -\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{64 d} \\ & = -\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{64 d} \\ & = -\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {\left (15 b^2 e\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+d x)\right )}{64 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\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+d x)\right )}{128 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e \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+d x)\right )}{128 d}-\frac {\left (15 b^2 e \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+d x)\right )}{128 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e \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+d x)}\right )}{64 d}-\frac {\left (15 b^2 e \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+d x)}\right )}{64 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^{5/2} e \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.55 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {i b^3 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )}{32 \sqrt {2} d \sqrt {a+b \arcsin (c+d x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(210)=420\).
Time = 0.85 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.75
method | result | size |
default | \(-\frac {e \left (15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}-15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}+32 \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -30 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+80 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+32 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-30 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b \right )}{128 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(449\) |
[In]
[Out]
Exception generated. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=e \left (\int a^{2} c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int b^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
\[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 1449, normalized size of antiderivative = 5.66 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
[In]
[Out]