Optimal. Leaf size=243 \[ -\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \text {ArcSin}(c+d x))^{7/2}}{d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 d} \]
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Rubi [A]
time = 0.28, 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 = {4887, 4715,
4767, 4719, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \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 \text {ArcSin}(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 \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \text {ArcSin}(c+d x))^{7/2}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4715
Rule 4719
Rule 4767
Rule 4887
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.38, size = 551, normalized size = 2.27 \begin {gather*} \frac {e^{-\frac {i a}{b}} \left (\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 {2 \pi } \sqrt {a+b \text {ArcSin}(c+d x)} \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right )-i \left (105 b^3 \left (-1+e^{\frac {2 i a}{b}}\right )+8 i a^3 \left (1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {2 \pi } \sqrt {a+b \text {ArcSin}(c+d x)} S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}\right )+\frac {4 \left (e^{\frac {i a}{b}} (a+b \text {ArcSin}(c+d x)) \left (7 \left (-10 a b (c+d x)+4 a^2 \sqrt {1-c^2-2 c d x-d^2 x^2}-15 b^2 \sqrt {1-c^2-2 c d x-d^2 x^2}\right )+\left (24 a^2 (c+d x)-70 b^2 (c+d x)+56 a b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \text {ArcSin}(c+d x)+4 b \left (6 a (c+d x)+7 b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \text {ArcSin}(c+d x)^2+8 b^2 (c+d x) \text {ArcSin}(c+d x)^3\right )+4 a^3 \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+4 a^3 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{\sqrt {\frac {1}{b}}}\right )}{32 \sqrt {\frac {1}{b}} d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(615\) vs.
\(2(197)=394\).
time = 0.21, size = 616, normalized size = 2.53
method | result | size |
default | \(-\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 {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, 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 {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b^{4}+16 \arcsin \left (d x +c \right )^{4} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}+64 \arcsin \left (d x +c \right )^{3} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}-56 \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}+96 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b^{2}-140 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}-168 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}+64 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{3} b -280 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}-168 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b^{2}+210 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\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 )}}\) | \(616\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.44, size = 2308, normalized size = 9.50 \begin {gather*} \text {Too large to display} \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 {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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