Optimal. Leaf size=653 \[ -\frac {48 i b \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}+\frac {48 i b \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}-\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}+\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d \sqrt {b^2-a^2}}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d \sqrt {b^2-a^2}}+\frac {2 x^{5/2}}{5 a} \]
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Rubi [A] time = 1.05, antiderivative size = 653, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {4204, 4191, 3321, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {8 b x^{3/2} \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {8 b x^{3/2} \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {24 i b x \text {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {24 i b x \text {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {48 b \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^4 \sqrt {b^2-a^2}}+\frac {48 b \sqrt {x} \text {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d^4 \sqrt {b^2-a^2}}-\frac {48 i b \text {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^5 \sqrt {b^2-a^2}}+\frac {48 i b \text {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d^5 \sqrt {b^2-a^2}}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d \sqrt {b^2-a^2}}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d \sqrt {b^2-a^2}}+\frac {2 x^{5/2}}{5 a} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3321
Rule 4191
Rule 4204
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{a+b \sec (c+d x)} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {x^4}{a}-\frac {b x^4}{a (b+a \cos (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{5/2}}{5 a}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^4}{b+a \cos (c+d x)} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {2 x^{5/2}}{5 a}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^4}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {2 x^{5/2}}{5 a}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^4}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^4}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}\\ &=\frac {2 x^{5/2}}{5 a}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {(8 i b) \operatorname {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}+\frac {(8 i b) \operatorname {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}\\ &=\frac {2 x^{5/2}}{5 a}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {(24 b) \operatorname {Subst}\left (\int x^2 \text {Li}_2\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(24 b) \operatorname {Subst}\left (\int x^2 \text {Li}_2\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}\\ &=\frac {2 x^{5/2}}{5 a}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(48 i b) \operatorname {Subst}\left (\int x \text {Li}_3\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}+\frac {(48 i b) \operatorname {Subst}\left (\int x \text {Li}_3\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}\\ &=\frac {2 x^{5/2}}{5 a}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {(48 b) \operatorname {Subst}\left (\int \text {Li}_4\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {(48 b) \operatorname {Subst}\left (\int \text {Li}_4\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}\\ &=\frac {2 x^{5/2}}{5 a}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {(48 i b) \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {(48 i b) \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^5}\\ &=\frac {2 x^{5/2}}{5 a}+\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^2 \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {8 b x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {24 i b x \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {48 b \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {48 i b \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {48 i b \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}\\ \end {align*}
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Mathematica [A] time = 2.49, size = 725, normalized size = 1.11 \[ \frac {2 \sec \left (c+d \sqrt {x}\right ) \left (a \cos \left (c+d \sqrt {x}\right )+b\right ) \left (x^{5/2}+\frac {5 i b e^{i c} \left (d^4 x^2 \log \left (1+\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {e^{2 i c} \left (b^2-a^2\right )}}\right )-d^4 x^2 \log \left (1+\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{\sqrt {e^{2 i c} \left (b^2-a^2\right )}+b e^{i c}}\right )-4 i d^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )+4 i d^3 x^{3/2} \text {Li}_2\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )+12 d^2 x \text {Li}_3\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )-12 d^2 x \text {Li}_3\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )+24 i d \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )-24 i d \sqrt {x} \text {Li}_4\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )-24 \text {Li}_5\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )+24 \text {Li}_5\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )\right )}{d^5 \sqrt {e^{2 i c} \left (b^2-a^2\right )}}\right )}{5 a \left (a+b \sec \left (c+d \sqrt {x}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{\frac {3}{2}}}{b \sec \left (d \sqrt {x} + c\right ) + a}, x\right ) \]
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 \[ \int \frac {x^{\frac {3}{2}}}{b \sec \left (d \sqrt {x} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {3}{2}}}{a +b \sec \left (c +d \sqrt {x}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^{3/2}}{a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}} \,d x \]
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 \[ \int \frac {x^{\frac {3}{2}}}{a + b \sec {\left (c + d \sqrt {x} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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