Optimal. Leaf size=274 \[ \frac {a^2 x^2}{2}-\frac {3 i a b \text {Li}_4\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {6 a b \sqrt {x} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 i a b x \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+i a b x^2+\frac {3 b^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{3/2}}{d}-\frac {1}{2} b^2 x^2 \]
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Rubi [A] time = 0.47, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3747, 3722, 3719, 2190, 2531, 6609, 2282, 6589, 3720, 30} \[ \frac {a^2 x^2}{2}+\frac {6 i a b x \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 a b \sqrt {x} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {3 i a b \text {Li}_4\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+i a b x^2-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {3 b^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{3/2}}{d}-\frac {1}{2} b^2 x^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3719
Rule 3720
Rule 3722
Rule 3747
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x \left (a+b \tan \left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \operatorname {Subst}\left (\int x^3 (a+b \tan (c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^2 x^3+2 a b x^3 \tan (c+d x)+b^2 x^3 \tan ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^2 x^2}{2}+(4 a b) \operatorname {Subst}\left (\int x^3 \tan (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \operatorname {Subst}\left (\int x^3 \tan ^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^2 x^2}{2}+i a b x^2+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-(8 i a b) \operatorname {Subst}\left (\int \frac {e^{2 i (c+d x)} x^3}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )-\left (2 b^2\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,\sqrt {x}\right )-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int x^2 \tan (c+d x) \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}+i a b x^2-\frac {b^2 x^2}{2}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(12 a b) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (12 i b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i (c+d x)} x^2}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}+i a b x^2-\frac {b^2 x^2}{2}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 i a b x \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(12 i a b) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}+i a b x^2-\frac {b^2 x^2}{2}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 i a b x \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 a b \sqrt {x} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(6 a b) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {\left (6 i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}\\ &=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}+i a b x^2-\frac {b^2 x^2}{2}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 i a b x \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 a b \sqrt {x} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(3 i a b) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}\\ &=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}+i a b x^2-\frac {b^2 x^2}{2}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 a b x^{3/2} \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {6 i b^2 \sqrt {x} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 i a b x \text {Li}_2\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {3 b^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {6 a b \sqrt {x} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {3 i a b \text {Li}_4\left (-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 2.32, size = 365, normalized size = 1.33 \[ \frac {1}{2} x^2 \left (a^2+2 a b \tan (c)-b^2\right )+\frac {b \left (-6 i \left (1+e^{2 i c}\right ) d \sqrt {x} \left (a d \sqrt {x}-b\right ) \text {Li}_2\left (-e^{-2 i \left (c+d \sqrt {x}\right )}\right )+3 \left (1+e^{2 i c}\right ) \left (b-2 a d \sqrt {x}\right ) \text {Li}_3\left (-e^{-2 i \left (c+d \sqrt {x}\right )}\right )-4 a e^{2 i c} d^3 x^{3/2} \log \left (1+e^{-2 i \left (c+d \sqrt {x}\right )}\right )-4 a d^3 x^{3/2} \log \left (1+e^{-2 i \left (c+d \sqrt {x}\right )}\right )+3 i a e^{2 i c} \text {Li}_4\left (-e^{-2 i \left (c+d \sqrt {x}\right )}\right )+3 i a \text {Li}_4\left (-e^{-2 i \left (c+d \sqrt {x}\right )}\right )-2 i a d^4 x^2+6 b e^{2 i c} d^2 x \log \left (1+e^{-2 i \left (c+d \sqrt {x}\right )}\right )+6 b d^2 x \log \left (1+e^{-2 i \left (c+d \sqrt {x}\right )}\right )+4 i b d^3 x^{3/2}\right )}{\left (1+e^{2 i c}\right ) d^4}+\frac {2 b^2 x^{3/2} \sec (c) \sin \left (d \sqrt {x}\right ) \sec \left (c+d \sqrt {x}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x \tan \left (d \sqrt {x} + c\right )^{2} + 2 \, a b x \tan \left (d \sqrt {x} + c\right ) + a^{2} x, x\right ) \]
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 \[ \int {\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}^{2} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.39, size = 0, normalized size = 0.00 \[ \int x \left (a +b \tan \left (c +d \sqrt {x}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.13, size = 1282, normalized size = 4.68 \[ \text {result too large to display} \]
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 \[ \int x\,{\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )}^2 \,d x \]
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 \[ \int x \left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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