Optimal. Leaf size=120 \[ \frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac {20 b d^{5/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{147 c^{7/2}}+\frac {4 b \sqrt {1-c^2 x^2} (d x)^{5/2}}{49 c}+\frac {20 b d^2 \sqrt {1-c^2 x^2} \sqrt {d x}}{147 c^3} \]
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Rubi [A] time = 0.08, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4627, 321, 329, 221} \[ \frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}+\frac {20 b d^2 \sqrt {1-c^2 x^2} \sqrt {d x}}{147 c^3}-\frac {20 b d^{5/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{147 c^{7/2}}+\frac {4 b \sqrt {1-c^2 x^2} (d x)^{5/2}}{49 c} \]
Antiderivative was successfully verified.
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Rule 221
Rule 321
Rule 329
Rule 4627
Rubi steps
\begin {align*} \int (d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac {(2 b c) \int \frac {(d x)^{7/2}}{\sqrt {1-c^2 x^2}} \, dx}{7 d}\\ &=\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac {(10 b d) \int \frac {(d x)^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{49 c}\\ &=\frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac {\left (10 b d^3\right ) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{147 c^3}\\ &=\frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac {\left (20 b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{147 c^3}\\ &=\frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 d}-\frac {20 b d^{5/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{147 c^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 100, normalized size = 0.83 \[ \frac {2 d^2 \sqrt {d x} \left (21 a c^3 x^3+21 b c^3 x^3 \sin ^{-1}(c x)-10 b \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )+6 b c^2 x^2 \sqrt {1-c^2 x^2}+10 b \sqrt {1-c^2 x^2}\right )}{147 c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b d^{2} x^{2} \arcsin \left (c x\right ) + a d^{2} x^{2}\right )} \sqrt {d x}, x\right ) \]
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 \[ \int \left (d x\right )^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 144, normalized size = 1.20 \[ \frac {\frac {2 \left (d x \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \arcsin \left (c x \right )}{7}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{7 c^{2}}-\frac {5 d^{4} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{21 c^{4}}+\frac {5 d^{4} \sqrt {-c x +1}\, \sqrt {c x +1}\, \EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{21 c^{4} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{7 d}\right )}{d} \]
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 \[ \frac {2}{7} \, b d^{\frac {5}{2}} x^{\frac {7}{2}} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + \frac {2}{7} \, {\left (a d^{2} x^{\frac {7}{2}} + 7 \, b c d^{2} \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{\frac {7}{2}}}{7 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x}\right )} \sqrt {d} \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 104.62, size = 82, normalized size = 0.68 \[ a \left (\begin {cases} 0 & \text {for}\: d = 0 \\\frac {2 \left (d x\right )^{\frac {7}{2}}}{7 d} & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} 0 & \text {for}\: d = 0 \\\frac {d^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {c^{2} x^{2} e^{2 i \pi }} \right )}}{7 \Gamma \left (\frac {13}{4}\right )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} 0 & \text {for}\: d = 0 \\\frac {2 \left (d x\right )^{\frac {7}{2}}}{7 d} & \text {otherwise} \end {cases}\right ) \operatorname {asin}{\left (c x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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