Optimal. Leaf size=287 \[ \frac{b^2 (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{5}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{b^2 (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{5}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (-b+i a) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}} \]
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Rubi [A] time = 0.358011, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3575, 912, 137, 136} \[ \frac{b^2 (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{5}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{b^2 (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{5}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (-b+i a) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3575
Rule 912
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^m}{(c+d x)^{5/2} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{i (a+b x)^m}{2 (i-x) (c+d x)^{5/2}}+\frac{i (a+b x)^m}{2 (i+x) (c+d x)^{5/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i-x) (c+d x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{i \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i+x) (c+d x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{\left (i b^2 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i-x) \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{2 (b c-a d)^2 f \sqrt{c+d \tan (e+f x)}}+\frac{\left (i b^2 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i+x) \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{2 (b c-a d)^2 f \sqrt{c+d \tan (e+f x)}}\\ &=\frac{b^2 F_1\left (1+m;\frac{5}{2},1;2+m;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a+b) (b c-a d)^2 f (1+m) \sqrt{c+d \tan (e+f x)}}-\frac{b^2 F_1\left (1+m;\frac{5}{2},1;2+m;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a-b) (b c-a d)^2 f (1+m) \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [F] time = 17.6258, size = 0, normalized size = 0. \[ \int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.339, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{m} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \tan \left (f x + e\right ) + c}{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d^{3} \tan \left (f x + e\right )^{3} + 3 \, c d^{2} \tan \left (f x + e\right )^{2} + 3 \, c^{2} d \tan \left (f x + e\right ) + c^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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