Optimal. Leaf size=256 \[ -\frac {15 \sqrt {\pi } b^{5/2} e \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 \sqrt {\pi } b^{5/2} e \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {15 b^2 e \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d} \]
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Rubi [A] time = 0.72, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4805, 12, 4629, 4707, 4641, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ -\frac {15 \sqrt {\pi } b^{5/2} e \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{128 d}-\frac {15 \sqrt {\pi } b^{5/2} e \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {15 b^2 e \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3351
Rule 3352
Rule 4629
Rule 4641
Rule 4707
Rule 4723
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {(5 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )^{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} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {(5 b e) \operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (15 b^2 e\right ) \operatorname {Subst}\left (\int x \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {\left (15 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {\left (15 b^3 e\right ) \operatorname {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {\left (15 b^3 e\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}\\ &=\frac {15 b^2 e \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e\right ) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{128 d}\\ &=\frac {15 b^2 e \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^3 e \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 )}{128 d}-\frac {\left (15 b^3 e \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 )}{128 d}\\ &=\frac {15 b^2 e \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {\left (15 b^2 e \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 )}{64 d}-\frac {\left (15 b^2 e \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 )}{64 d}\\ &=\frac {15 b^2 e \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac {15 b^{5/2} e \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 )}{128 d}-\frac {15 b^{5/2} e \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 )}{128 d}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 154, normalized size = 0.60 \[ \frac {e e^{-\frac {2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \left (\sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {7}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{32 \sqrt {2} d \left (\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
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 [B] time = 2.32, size = 1534, normalized size = 5.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 435, normalized size = 1.70 \[ -\frac {e \left (15 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{3}+15 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{3}+32 \arcsin \left (d x +c \right )^{3} \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a \,b^{2}-40 \arcsin \left (d x +c \right )^{2} \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a^{2} b -30 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{3}-80 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a \,b^{2}+32 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a^{3}-30 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a \,b^{2}-40 \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a^{2} b \right )}{128 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e \left (\int a^{2} c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int b^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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