Optimal. Leaf size=264 \[ \frac {d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}} \]
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Rubi [A] time = 2.50, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3566, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac {d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 3566
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2}}{\sqrt {a+b \tan (e+f x)}} \, dx &=\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\int \frac {\frac {1}{2} \left (2 b c^3-d^2 (b c+a d)\right )+b d \left (3 c^2-d^2\right ) \tan (e+f x)+\frac {1}{2} d^2 (5 b c-a d) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{b}\\ &=\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (2 b c^3-d^2 (b c+a d)\right )+b d \left (3 c^2-d^2\right ) x+\frac {1}{2} d^2 (5 b c-a d) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b f}\\ &=\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\operatorname {Subst}\left (\int \left (\frac {d^2 (5 b c-a d)}{2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {b c \left (c^2-3 d^2\right )+b d \left (3 c^2-d^2\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{b f}\\ &=\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\operatorname {Subst}\left (\int \frac {b c \left (c^2-3 d^2\right )+b d \left (3 c^2-d^2\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b f}+\frac {\left (d^2 (5 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b f}\\ &=\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\operatorname {Subst}\left (\int \left (\frac {i b c \left (c^2-3 d^2\right )-b d \left (3 c^2-d^2\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {i b c \left (c^2-3 d^2\right )+b d \left (3 c^2-d^2\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b f}+\frac {\left (d^2 (5 b c-a d)\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 )}{b^2 f}\\ &=\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {(i c-d)^3 \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 {(i c+d)^3 \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^2 (5 b c-a d)\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 )}{b^2 f}\\ &=\frac {d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {(i c-d)^3 \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 c+d)^3 \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 (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 )}{\sqrt {a-i b} 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 )}{\sqrt {a+i b} f}+\frac {d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{3/2} f}+\frac {d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}\\ \end {align*}
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Mathematica [A] time = 2.88, size = 432, normalized size = 1.64 \[ \frac {\frac {b \left (\sqrt {-b^2} c \left (c^2-3 d^2\right )-b d \left (d^2-3 c^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}-c} \sqrt {a+b \tan (e+f x)}}{\sqrt {\sqrt {-b^2}-a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {\sqrt {-b^2}-a} \sqrt {\frac {\sqrt {-b^2} d}{b}-c}}+\frac {b \left (\sqrt {-b^2} c \left (c^2-3 d^2\right )+b d \left (d^2-3 c^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}+c} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {\frac {\sqrt {-b^2} d}{b}+c}}+\frac {\sqrt {b} d^{3/2} (5 b c-a d) \sqrt {c-\frac {a d}{b}} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right )}{\sqrt {c+d \tan (e+f x)}}+b d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^2 f} \]
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 \[ \int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{\sqrt {a +b \tan \left (f x +e \right )}}\, dx \]
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 \[ \int \frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {b \tan \left (f x + e\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{\sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\sqrt {a + b \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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