Optimal. Leaf size=429 \[ -\frac {i (a-i b)^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i (a+i b)^{3/2} (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt {d} f}+\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f} \]
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Rubi [A]
time = 4.09, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3647, 3728,
3736, 6857, 65, 223, 212, 95, 214} \begin {gather*} \frac {\left (-a^2 d^2+14 a b c d+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {\left (-a^3 d^3+15 a^2 b c d^2+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt {d} f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}-\frac {i (a-i b)^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i (a+i b)^{3/2} (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3647
Rule 3728
Rule 3736
Rule 6857
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx &=\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (\frac {1}{2} \left (6 b c^3-5 b c d^2-a d^3\right )+3 b d \left (3 c^2-d^2\right ) \tan (e+f x)+\frac {1}{2} d^2 (13 b c-a d) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 b}\\ &=\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (-\frac {3}{4} d \left (13 b^2 c^2 d+a^2 d^3-a b \left (8 c^3-10 c d^2\right )\right )+6 b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) \tan (e+f x)+\frac {3}{4} d^2 \left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{6 b d}\\ &=\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\int \frac {-\frac {3}{8} d^2 \left (a^3 d^3+b^3 c \left (11 c^2-8 d^2\right )+a b^2 d \left (51 c^2-8 d^2\right )-a^2 b \left (16 c^3-33 c d^2\right )\right )+6 b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)+\frac {3}{8} d^2 \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{6 b d^2}\\ &=\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{8} d^2 \left (a^3 d^3+b^3 c \left (11 c^2-8 d^2\right )+a b^2 d \left (51 c^2-8 d^2\right )-a^2 b \left (16 c^3-33 c d^2\right )\right )+6 b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) x+\frac {3}{8} d^2 \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 b d^2 f}\\ &=\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\text {Subst}\left (\int \left (\frac {3 d^2 \left (15 a^2 b c d^2-a^3 d^3+5 b^3 \left (c^3-8 c d^2\right )+a b^2 \left (45 c^2 d-24 d^3\right )\right )}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {6 \left (-b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )+b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{6 b d^2 f}\\ &=\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\text {Subst}\left (\int \frac {-b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )+b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b d^2 f}+\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{16 b f}\\ &=\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\text {Subst}\left (\int \left (\frac {-i b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )-b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {-i b d^2 \left (b^2 c \left (c^2-3 d^2\right )+2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right )+b d^2 \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b d^2 f}+\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{8 b^2 f}\\ &=\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}-\frac {\left ((a+i b)^2 (i c-d)^3\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}-\frac {\left ((a-i b)^2 (i c+d)^3\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{8 b^2 f}\\ &=\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt {d} f}+\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}-\frac {\left ((a+i b)^2 (i c-d)^3\right ) \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left ((a-i b)^2 (i c+d)^3\right ) \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}\\ &=-\frac {i (a-i b)^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i (a+i b)^{3/2} (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} \sqrt {d} f}+\frac {\left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b f}+\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 b f}+\frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}\\ \end {align*}
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Mathematica [A]
time = 8.21, size = 773, normalized size = 1.80 \begin {gather*} \frac {d^2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 b f}+\frac {\frac {d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {\frac {3 d \left (14 a b c d-a^2 d^2+b^2 \left (11 c^2-8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {-\frac {6 b d^2 \left (\sqrt {-b^2} \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right )-b \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {6 b d^2 \left (\sqrt {-b^2} \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )+a b \left (6 c^2 d-2 d^3\right )\right )+b \left (2 a b c \left (c^2-3 d^2\right )-b^2 d \left (3 c^2-d^2\right )+a^2 \left (3 c^2 d-d^3\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+\frac {3 \sqrt {b} d^{3/2} \sqrt {c-\frac {a d}{b}} \left (15 a^2 b c d^2-a^3 d^3+3 a b^2 d \left (15 c^2-8 d^2\right )+5 b^3 \left (c^3-8 c d^2\right )\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt {c+d \tan (e+f x)}}}{b d f}}{2 d}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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