Optimal. Leaf size=86 \[ \frac {10 b x \sqrt {1-c^2 x^4}}{147 c^3}+\frac {2 b x^5 \sqrt {1-c^2 x^4}}{49 c}+\frac {1}{7} x^7 \left (a+b \text {ArcSin}\left (c x^2\right )\right )-\frac {10 b F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{147 c^{7/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4926, 12, 327,
227} \begin {gather*} \frac {1}{7} x^7 \left (a+b \text {ArcSin}\left (c x^2\right )\right )-\frac {10 b F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{147 c^{7/2}}+\frac {2 b x^5 \sqrt {1-c^2 x^4}}{49 c}+\frac {10 b x \sqrt {1-c^2 x^4}}{147 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 227
Rule 327
Rule 4926
Rubi steps
\begin {align*} \int x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{7} b \int \frac {2 c x^8}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{7} (2 b c) \int \frac {x^8}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {2 b x^5 \sqrt {1-c^2 x^4}}{49 c}+\frac {1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {(10 b) \int \frac {x^4}{\sqrt {1-c^2 x^4}} \, dx}{49 c}\\ &=\frac {10 b x \sqrt {1-c^2 x^4}}{147 c^3}+\frac {2 b x^5 \sqrt {1-c^2 x^4}}{49 c}+\frac {1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {(10 b) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx}{147 c^3}\\ &=\frac {10 b x \sqrt {1-c^2 x^4}}{147 c^3}+\frac {2 b x^5 \sqrt {1-c^2 x^4}}{49 c}+\frac {1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {10 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{147 c^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 82, normalized size = 0.95 \begin {gather*} \frac {1}{147} \left (21 a x^7+\frac {2 b x \sqrt {1-c^2 x^4} \left (5+3 c^2 x^4\right )}{c^3}+21 b x^7 \text {ArcSin}\left (c x^2\right )-\frac {10 i b F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )}{(-c)^{7/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 108, normalized size = 1.26
method | result | size |
default | \(\frac {x^{7} a}{7}+b \left (\frac {x^{7} \arcsin \left (c \,x^{2}\right )}{7}-\frac {2 c \left (-\frac {x^{5} \sqrt {-c^{2} x^{4}+1}}{7 c^{2}}-\frac {5 x \sqrt {-c^{2} x^{4}+1}}{21 c^{4}}+\frac {5 \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{21 c^{\frac {9}{2}} \sqrt {-c^{2} x^{4}+1}}\right )}{7}\right )\) | \(108\) |
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 [A]
time = 0.81, size = 58, normalized size = 0.67 \begin {gather*} \frac {21 \, b c^{3} x^{7} \arcsin \left (c x^{2}\right ) + 21 \, a c^{3} x^{7} + 2 \, {\left (3 \, b c^{2} x^{5} + 5 \, b x\right )} \sqrt {-c^{2} x^{4} + 1}}{147 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.66, size = 58, normalized size = 0.67 \begin {gather*} \frac {a x^{7}}{7} - \frac {b c x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{14 \Gamma \left (\frac {13}{4}\right )} + \frac {b x^{7} \operatorname {asin}{\left (c x^{2} \right )}}{7} \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 = -0.01 \begin {gather*} \int x^6\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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