Optimal. Leaf size=468 \[ \frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 (a+b \text {ArcSin}(c+d x))^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 (a+b \text {ArcSin}(c+d x))^{5/2}}-\frac {4}{15 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {8 c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 c \sqrt {2 \pi } \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d^2} \]
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Rubi [A]
time = 0.71, 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 = {4889, 4829, 4717, 4807, 4809, 3387, 3386, 3432, 3385, 3433, 4729, 4727, 4737}
\begin {gather*} \frac {8 \sqrt {2 \pi } c \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 )}{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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(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 \text {ArcSin}(c+d x)}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {8 (c+d x)^2}{15 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {4}{15 b^2 d^2 (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b d^2 (a+b \text {ArcSin}(c+d x))^{5/2}}+\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4717
Rule 4727
Rule 4729
Rule 4737
Rule 4807
Rule 4809
Rule 4829
Rule 4889
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sin ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\text {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 {\text {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 {\text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {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 \text {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 \text {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) \text {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 \text {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) \text {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 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 1.22, size = 524, normalized size = 1.12 \begin {gather*} \frac {-c \left (-6 b^2 e^{i \text {ArcSin}(c+d x)}+4 e^{-\frac {i a}{b}} (a+b \text {ArcSin}(c+d x)) \left (e^{\frac {i (a+b \text {ArcSin}(c+d x))}{b}} (2 a-i b+2 b \text {ArcSin}(c+d x))-2 i b \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )+e^{-i \text {ArcSin}(c+d x)} \left (8 a^2+4 a b (i+4 \text {ArcSin}(c+d x))+2 b^2 \left (-3+2 i \text {ArcSin}(c+d x)+4 \text {ArcSin}(c+d x)^2\right )-8 e^{\frac {i (a+b \text {ArcSin}(c+d x))}{b}} (a+b \text {ArcSin}(c+d x))^2 \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 )\right )\right )-2 \left (4 a b \cos (2 \text {ArcSin}(c+d x))+4 b^2 \text {ArcSin}(c+d x) \cos (2 \text {ArcSin}(c+d x))+32 \sqrt {\frac {1}{b}} \sqrt {\pi } (a+b \text {ArcSin}(c+d x))^{5/2} \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right )+32 \sqrt {\frac {1}{b}} \sqrt {\pi } (a+b \text {ArcSin}(c+d x))^{5/2} S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )-16 a^2 \sin (2 \text {ArcSin}(c+d x))+3 b^2 \sin (2 \text {ArcSin}(c+d x))-32 a b \text {ArcSin}(c+d x) \sin (2 \text {ArcSin}(c+d x))-16 b^2 \text {ArcSin}(c+d x)^2 \sin (2 \text {ArcSin}(c+d x))\right )}{30 b^3 d^2 (a+b \text {ArcSin}(c+d x))^{5/2}} \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. \(1255\) vs.
\(2(384)=768\).
time = 0.52, size = 1256, normalized size = 2.68
method | result | size |
default | \(\text {Expression too large to display}\) | \(1256\) |
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*} \int \frac {x}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \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 {x}{{\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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