Integrand size = 27, antiderivative size = 283 \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\frac {(b c-a d) \operatorname {AppellF1}\left (1+m,-\frac {3}{2},1,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt {c+d \tan (e+f x)}}{2 b (i a+b) f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {(b c-a d) \operatorname {AppellF1}\left (1+m,-\frac {3}{2},1,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt {c+d \tan (e+f x)}}{2 (i a-b) b f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}} \]
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Time = 0.60 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3656, 926, 142, 141} \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\frac {(b c-a d) \sqrt {c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {3}{2},1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 b f (m+1) (b+i a) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {3}{2},1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 b f (m+1) (-b+i a) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}} \]
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Rule 141
Rule 142
Rule 926
Rule 3656
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^m (c+d x)^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {i (a+b x)^m (c+d x)^{3/2}}{2 (i-x)}+\frac {i (a+b x)^m (c+d x)^{3/2}}{2 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {i \text {Subst}\left (\int \frac {(a+b x)^m (c+d x)^{3/2}}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {i \text {Subst}\left (\int \frac {(a+b x)^m (c+d x)^{3/2}}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {\left (i (b c-a d) \sqrt {c+d \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{3/2}}{i-x} \, dx,x,\tan (e+f x)\right )}{2 b f \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {\left (i (b c-a d) \sqrt {c+d \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{3/2}}{i+x} \, dx,x,\tan (e+f x)\right )}{2 b f \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}} \\ & = \frac {(b c-a d) \operatorname {AppellF1}\left (1+m,-\frac {3}{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 {c+d \tan (e+f x)}}{2 b (i a+b) f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {(b c-a d) \operatorname {AppellF1}\left (1+m,-\frac {3}{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 {c+d \tan (e+f x)}}{2 (i a-b) b f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}} \\ \end{align*}
\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx \]
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\[\int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2} \,d x \]
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