Optimal. Leaf size=468 \[ \frac {8 \sqrt {2 \pi } c \sin \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {8 \sqrt {2 \pi } c \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {32 \sqrt {1-(c+d x)^2} (c+d x)}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}+\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 1.06, antiderivative size = 468, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {4805, 4745, 4621, 4719, 4723, 3306, 3305, 3351, 3304, 3352, 4633, 4631, 4641} \[ \frac {8 \sqrt {2 \pi } c \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {8 \sqrt {2 \pi } c \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {32 \sqrt {1-(c+d x)^2} (c+d x)}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}+\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4621
Rule 4631
Rule 4633
Rule 4641
Rule 4719
Rule 4723
Rule 4745
Rule 4805
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sin ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {c}{d \left (a+b \sin ^{-1}(x)\right )^{7/2}}+\frac {x}{d \left (a+b \sin ^{-1}(x)\right )^{7/2}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}-\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 \operatorname {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d^2}+\frac {(4 c) \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {32 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac {(8 c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {(8 c) \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac {\left (32 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac {\left (32 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (8 c \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 )}{15 b^3 d^2}-\frac {\left (64 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d^2}+\frac {\left (8 c \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 )}{15 b^3 d^2}-\frac {\left (64 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d^2}-\frac {\left (16 c \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 )}{15 b^4 d^2}+\frac {\left (16 c \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 )}{15 b^4 d^2}\\ &=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {8 c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 c \sqrt {2 \pi } C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d^2}\\ \end {align*}
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Mathematica [C] time = 2.01, size = 524, normalized size = 1.12 \[ \frac {-2 \left (-16 a^2 \sin \left (2 \sin ^{-1}(c+d x)\right )+32 \sqrt {\pi } \sqrt {\frac {1}{b}} \cos \left (\frac {2 a}{b}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )+32 \sqrt {\pi } \sqrt {\frac {1}{b}} \sin \left (\frac {2 a}{b}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi }}\right )-32 a b \sin ^{-1}(c+d x) \sin \left (2 \sin ^{-1}(c+d x)\right )+4 a b \cos \left (2 \sin ^{-1}(c+d x)\right )+3 b^2 \sin \left (2 \sin ^{-1}(c+d x)\right )-16 b^2 \sin ^{-1}(c+d x)^2 \sin \left (2 \sin ^{-1}(c+d x)\right )+4 b^2 \sin ^{-1}(c+d x) \cos \left (2 \sin ^{-1}(c+d x)\right )\right )-c \left (e^{-i \sin ^{-1}(c+d x)} \left (8 a^2+4 a b \left (4 \sin ^{-1}(c+d x)+i\right )-8 e^{\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^2 \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 b^2 \left (4 \sin ^{-1}(c+d x)^2+2 i \sin ^{-1}(c+d x)-3\right )\right )+4 e^{-\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right ) \left (e^{\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \left (2 a+2 b \sin ^{-1}(c+d x)-i b\right )-2 i b \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )-6 b^2 e^{i \sin ^{-1}(c+d x)}\right )}{30 b^3 d^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 1172, normalized size = 2.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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