Optimal. Leaf size=322 \[ \frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 \left (2 a^3 c d+3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {(b+i a)^3 (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(-b+i a)^3 (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f} \]
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Rubi [A] time = 0.91, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3566, 3630, 3528, 3539, 3537, 63, 208} \[ \frac {2 \left (3 a^2 b \left (c^2-d^2\right )+2 a^3 c d-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 \left (3 a^2 b c+a^3 d-3 a b^2 d-b^3 c\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {(b+i a)^3 (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(-b+i a)^3 (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 3528
Rule 3537
Rule 3539
Rule 3566
Rule 3630
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2} \, dx &=\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \tan (e+f x))^{5/2} \left (\frac {1}{2} \left (9 a^3 d-2 b^2 \left (b c+\frac {7 a d}{2}\right )\right )+\frac {9}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (b c-10 a d) \tan ^2(e+f x)\right ) \, dx}{9 d}\\ &=-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \tan (e+f x))^{5/2} \left (\frac {9}{2} a \left (a^2-3 b^2\right ) d+\frac {9}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {9}{2} d \left (a^3 c-3 a b^2 c-3 a^2 b d+b^3 d\right )+\frac {9}{2} d \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (-\frac {9}{2} d \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right )+\frac {9}{2} d \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int \frac {\frac {9}{2} d (a c-b d) \left (a^2 c^2-3 b^2 c^2-8 a b c d-3 a^2 d^2+b^2 d^2\right )+\frac {9}{2} d (b c+a d) \left (3 a^2 c^2-b^2 c^2-8 a b c d-a^2 d^2+3 b^2 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{9 d}\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {1}{2} \left ((a-i b)^3 (c-i d)^3\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (c+i d)^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac {\left ((i a+b)^3 (c-i d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}+\frac {\left ((i a-b)^3 (c+i d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac {\left ((a-i b)^3 (c-i d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b)^3 (c+i d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=\frac {(i a+b)^3 (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}\\ \end {align*}
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Mathematica [A] time = 6.20, size = 413, normalized size = 1.28 \[ \frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \left (\frac {i \left (\frac {9}{2} a d \left (a^2-3 b^2\right )-\frac {9}{2} i b d \left (3 a^2-b^2\right )\right ) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+(c-i d) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (2 \sqrt {c+d \tan (e+f x)}+\frac {2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}\right )\right )\right )}{2 f}-\frac {i \left (\frac {9}{2} a d \left (a^2-3 b^2\right )+\frac {9}{2} i b d \left (3 a^2-b^2\right )\right ) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+(c+i d) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (2 \sqrt {c+d \tan (e+f x)}+\frac {2 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}\right )\right )\right )}{2 f}-\frac {2 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{7 d f}\right )}{9 d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 5053, normalized size = 15.69 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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