Optimal. Leaf size=339 \[ \frac {\sqrt {d} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}-\frac {i \sqrt {a-i b} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f} \]
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Rubi [A] time = 3.78, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3566, 3647, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac {\sqrt {d} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}-\frac {i \sqrt {a-i b} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 3566
Rule 3647
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx &=\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (\frac {1}{2} \left (4 b c^3-3 b c d^2-a d^3\right )+2 b d \left (3 c^2-d^2\right ) \tan (e+f x)+\frac {1}{2} d^2 (9 b c-a d) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 b}\\ &=\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\int \frac {-\frac {1}{4} d \left (9 b^2 c^2 d+a^2 d^3-a b \left (8 c^3-14 c d^2\right )\right )+2 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 {1}{4} d^2 \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 b d}\\ &=\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} d \left (9 b^2 c^2 d+a^2 d^3-a b \left (8 c^3-14 c d^2\right )\right )+2 b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) x+\frac {1}{4} d^2 \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 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 )}{2 b d f}\\ &=\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\operatorname {Subst}\left (\int \left (\frac {d^2 \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )}{4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 \left (-b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )+b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{2 b d f}\\ &=\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\operatorname {Subst}\left (\int \frac {-b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )+b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b d f}+\frac {\left (d \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b f}\\ &=\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-i b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {b d \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-i b d \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b d f}+\frac {\left (d \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )\right ) \operatorname {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 )}{4 b^2 f}\\ &=\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\left ((i a+b) (c-i d)^3\right ) \operatorname {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 ((i a-b) (c+i d)^3\right ) \operatorname {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 (d \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right )\right ) \operatorname {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 )}{4 b^2 f}\\ &=\frac {\sqrt {d} \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 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 )}{4 b^{3/2} f}+\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\left ((i a+b) (c-i d)^3\right ) \operatorname {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 ((i a-b) (c+i d)^3\right ) \operatorname {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 \sqrt {a-i b} (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 \sqrt {a+i b} (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 {\sqrt {d} \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 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 )}{4 b^{3/2} f}+\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\\ \end {align*}
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Mathematica [A] time = 6.60, size = 550, normalized size = 1.62 \[ \frac {\frac {\sqrt {d} \sqrt {c-\frac {a d}{b}} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right )}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}+\frac {4 \left (b d \left (\sqrt {-b^2}-a\right ) \left (d^2-3 c^2\right )+a \sqrt {-b^2} c \left (c^2-3 d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}-c} \sqrt {a+b \tan (e+f x)}}{\sqrt {\sqrt {-b^2}-a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {\sqrt {-b^2}-a} \sqrt {\frac {\sqrt {-b^2} d}{b}-c}}-\frac {4 \left (-b d \left (a+\sqrt {-b^2}\right ) \left (d^2-3 c^2\right )-a \sqrt {-b^2} c \left (c^2-3 d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}+c} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {\frac {\sqrt {-b^2} d}{b}+c}}+2 d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}+d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f} \]
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 \[ \int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int \sqrt {b \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int \sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \]
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 \[ \int \sqrt {a + b \tan {\left (e + f x \right )}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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