Optimal. Leaf size=243 \[ \frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4803, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352} \[ \frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4619
Rule 4623
Rule 4677
Rule 4803
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}-\frac {(7 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}-\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}+\frac {\left (105 b^3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3 \cos \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 )}{8 d}+\frac {\left (105 b^3 \sin \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 )}{8 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.29, size = 551, normalized size = 2.27 \[ \frac {e^{-\frac {i a}{b}} \left (\sqrt {2 \pi } \left (8 i a^3 \left (-1+e^{\frac {2 i a}{b}}\right )+105 b^3 \left (1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {a+b \sin ^{-1}(c+d x)} C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-i \sqrt {2 \pi } \left (8 i a^3 \left (1+e^{\frac {2 i a}{b}}\right )+105 b^3 \left (-1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {a+b \sin ^{-1}(c+d x)} S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+\frac {4 \left (4 a^3 \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+4 a^3 e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right ) \left (7 \left (4 a^2 \sqrt {-c^2-2 c d x-d^2 x^2+1}-10 a b (c+d x)-15 b^2 \sqrt {-c^2-2 c d x-d^2 x^2+1}\right )+\sin ^{-1}(c+d x) \left (24 a^2 (c+d x)+56 a b \sqrt {-c^2-2 c d x-d^2 x^2+1}-70 b^2 (c+d x)\right )+4 b \sin ^{-1}(c+d x)^2 \left (6 a (c+d x)+7 b \sqrt {-c^2-2 c d x-d^2 x^2+1}\right )+8 b^2 (c+d x) \sin ^{-1}(c+d x)^3\right )\right )}{\sqrt {\frac {1}{b}}}\right )}{32 \sqrt {\frac {1}{b}} d \sqrt {a+b \sin ^{-1}(c+d x)}} \]
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 = 5.07, size = 2541, normalized size = 10.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 608, normalized size = 2.50 \[ \frac {105 \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, b^{4}+105 \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, b^{4}+16 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{4} b^{4}+64 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} a \,b^{3}+56 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} b^{4}+96 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a^{2} b^{2}-140 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} b^{4}+168 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a \,b^{3}+64 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{3} b -280 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a \,b^{3}+168 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{2} b^{2}-210 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) b^{4}+16 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a^{4}-140 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a^{2} b^{2}+56 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a^{3} b -210 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) a \,b^{3}}{16 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x \]
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]
________________________________________________________________________________________