Optimal. Leaf size=427 \[ -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^2 \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} e^2 \sin \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{144 d}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} e^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{144 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.33, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {4805, 12, 4629, 4707, 4677, 4619, 4723, 3306, 3305, 3351, 3304, 3352, 3312} \[ -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} e^2 \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{144 d}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} e^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{144 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3351
Rule 3352
Rule 4619
Rule 4629
Rule 4677
Rule 4707
Rule 4723
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int x^2 \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{6 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{72 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {a+b x}}-\frac {\sin (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{72 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}+\frac {\left (5 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}-\frac {\left (5 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b^2 e^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{6 d}+\frac {\left (5 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}-\frac {\left (5 b^3 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}-\frac {\left (5 b^2 e^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{6 d}-\frac {\left (5 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}+\frac {\left (5 b^3 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 d}-\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{6 d}+\frac {\left (5 b^2 e^2 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{48 d}-\frac {\left (5 b^2 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{144 d}-\frac {\left (5 b^2 e^2 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{48 d}+\frac {\left (5 b^2 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{144 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {15 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{144 d}-\frac {15 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 d}+\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.30, size = 249, normalized size = 0.58 \[ \frac {b^3 e^2 e^{-\frac {3 i a}{b}} \left (-81 e^{\frac {2 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {7}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-81 e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {7}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt {3} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {7}{2},-\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {7}{2},\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{648 d \sqrt {a+b \sin ^{-1}(c+d x)}} \]
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 [B] time = 5.98, size = 3080, normalized size = 7.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.43, size = 873, normalized size = 2.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{2} \left (\int a^{2} c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________