Optimal. Leaf size=398 \[ -\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {2 \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^3 \sqrt {a+b \tan (e+f x)}}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]
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Rubi [A] time = 2.36, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3565, 3649, 3616, 3615, 93, 208} \[ \frac {2 \left (-3 a^2 b^2 \left (15 c^2-13 d^2\right )+20 a^3 b c d+2 a^4 d^2-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^3 \sqrt {a+b \tan (e+f x)}}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 208
Rule 3565
Rule 3615
Rule 3616
Rule 3649
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx &=-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {2 \int \frac {\frac {1}{2} \left (11 b^2 c^2 d+a^2 d^3+a b c \left (5 c^2-7 d^2\right )\right )-\frac {5}{2} b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) \tan (e+f x)+\frac {1}{2} d \left (\left (a^2+5 b^2\right ) d^2-4 b c (b c-2 a d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}} \, dx}{5 b \left (a^2+b^2\right )}\\ &=-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {4 \int \frac {-\frac {1}{4} (b c-a d) \left (15 a^2 b c^3-15 b^3 c^3+70 a b^2 c^2 d-27 a^2 b c d^2+23 b^3 c d^2+2 a^3 d^3-8 a b^2 d^3\right )+\frac {15}{4} b (b c-a d) \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 {1}{2} d (b c-a d)^2 \left (10 a b c+a^2 d+11 b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}} \, dx}{15 b \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}+\frac {8 \int \frac {\frac {15}{8} b (b c-a d)^2 (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 {15}{8} b (b c-a d)^3 \left (8 a b c d-b^2 \left (c^2-3 d^2\right )+a^2 \left (3 c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2}\\ &=-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}+\frac {(c-i d)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac {(c+i d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}\\ &=-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}+\frac {(c-i d)^3 \operatorname {Subst}\left (\int \frac {1}{(1-i x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^3 f}+\frac {(c+i d)^3 \operatorname {Subst}\left (\int \frac {1}{(1+i x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^3 f}\\ &=-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}+\frac {(c-i d)^3 \operatorname {Subst}\left (\int \frac {1}{i a+b-(i c+d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^3 f}+\frac {(c+i d)^3 \operatorname {Subst}\left (\int \frac {1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^3 f}\\ &=-\frac {i (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 )}{(a-i b)^{7/2} f}+\frac {i (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 )}{(a+i b)^{7/2} f}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.26, size = 490, normalized size = 1.23 \[ \frac {(c+d \tan (e+f x))^{5/2}}{5 f (-b+i a) (a+b \tan (e+f x))^{5/2}}-\frac {(c+d \tan (e+f x))^{5/2}}{5 f (b+i a) (a+b \tan (e+f x))^{5/2}}+\frac {(d+i c) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a-i b) (a+b \tan (e+f x))^{3/2}}+\frac {3 (c-i d) \left (\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}+\frac {\sqrt {-c+i d} \tanh ^{-1}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}\right )}{a-i b}\right )}{3 f (a-i b)}-\frac {(-d+i c) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a+i b) (a+b \tan (e+f x))^{3/2}}-\frac {3 (c+i d) \left (\frac {\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2}}-\frac {\sqrt {c+d \tan (e+f x)}}{(a+i b) \sqrt {a+b \tan (e+f x)}}\right )}{a+i b}\right )}{3 f (a+i b)} \]
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 [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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