Optimal. Leaf size=412 \[ -\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {e^4 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{2 b^{3/2} d}+\frac {3 e^4 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {e^4 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {e^4 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 b^{3/2} d}-\frac {3 e^4 \sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^{3/2} d}+\frac {e^4 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} d} \]
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Rubi [A]
time = 0.57, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4889, 12,
4727, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} e^4 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{2 b^{3/2} d}-\frac {3 \sqrt {\frac {3 \pi }{2}} e^4 \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {\sqrt {\frac {5 \pi }{2}} e^4 \sin \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {\sqrt {\frac {\pi }{2}} e^4 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{2 b^{3/2} d}+\frac {3 \sqrt {\frac {3 \pi }{2}} e^4 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {\sqrt {\frac {5 \pi }{2}} e^4 \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4727
Rule 4889
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^4}{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int \frac {x^4}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {\left (2 e^4\right ) \text {Subst}\left (\int \left (-\frac {\sin (x)}{8 \sqrt {a+b x}}+\frac {9 \sin (3 x)}{16 \sqrt {a+b x}}-\frac {5 \sin (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {e^4 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 b d}-\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}+\frac {\left (9 e^4\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}\\ &=-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 b d}+\frac {\left (9 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \text {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 )}{8 b d}-\frac {\left (5 e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 b d}-\frac {\left (9 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \text {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 )}{8 b d}+\frac {\left (5 e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b d}\\ &=-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2 b^2 d}+\frac {\left (9 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}-\frac {\left (5 e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2 b^2 d}-\frac {\left (9 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}+\frac {\left (5 e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{4 b^2 d}\\ &=-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {e^4 \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 )}{2 b^{3/2} d}+\frac {3 e^4 \sqrt {\frac {3 \pi }{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 )}{4 b^{3/2} d}-\frac {e^4 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {e^4 \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 )}{2 b^{3/2} d}-\frac {3 e^4 \sqrt {\frac {3 \pi }{2}} 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 )}{4 b^{3/2} d}+\frac {e^4 \sqrt {\frac {5 \pi }{2}} C\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.42, size = 572, normalized size = 1.39 \begin {gather*} \frac {e^4 e^{-\frac {5 i (a+b \text {ArcSin}(c+d x))}{b}} \left (-e^{\frac {5 i a}{b}}+3 e^{\frac {5 i a}{b}+2 i \text {ArcSin}(c+d x)}-2 e^{\frac {5 i a}{b}+4 i \text {ArcSin}(c+d x)}-2 e^{\frac {5 i a}{b}+6 i \text {ArcSin}(c+d x)}+3 e^{\frac {5 i a}{b}+8 i \text {ArcSin}(c+d x)}-e^{\frac {5 i (a+2 b \text {ArcSin}(c+d x))}{b}}+2 e^{\frac {4 i a}{b}+5 i \text {ArcSin}(c+d x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+2 e^{\frac {6 i a}{b}+5 i \text {ArcSin}(c+d x)} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )-3 \sqrt {3} e^{\frac {2 i a}{b}+5 i \text {ArcSin}(c+d x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )-3 \sqrt {3} e^{\frac {8 i a}{b}+5 i \text {ArcSin}(c+d x)} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {5} e^{5 i \text {ArcSin}(c+d x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {5 i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {5} e^{\frac {5 i (2 a+b \text {ArcSin}(c+d x))}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {5 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{16 b d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 483, normalized size = 1.17
method | result | size |
default | \(\frac {e^{4} \left (2 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}+2 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}-3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {\pi }-3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {\pi }+\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {5 a}{b}\right ) \mathrm {S}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {-\frac {5}{b}}\, \sqrt {\pi }+\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {5 a}{b}\right ) \FresnelC \left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {-\frac {5}{b}}\, \sqrt {\pi }+3 \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right )-2 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right )-\cos \left (-\frac {5 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {5 a}{b}\right )\right )}{8 d b \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(483\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} e^{4} \left (\int \frac {c^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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