Optimal. Leaf size=283 \[ \frac {b F_1\left (1+m;\frac {3}{2},1;2+m;-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a+b) (b c-a d) f (1+m) \sqrt {c+d \tan (e+f x)}}-\frac {b F_1\left (1+m;\frac {3}{2},1;2+m;-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a-b) (b c-a d) f (1+m) \sqrt {c+d \tan (e+f x)}} \]
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Rubi [A]
time = 0.25, antiderivative size = 283, normalized size of antiderivative = 1.00, 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 = {3656, 926, 142,
141} \begin {gather*} \frac {b (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 {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 f (m+1) (b+i a) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {b (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 {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 f (m+1) (-b+i a) (b c-a d) \sqrt {c+d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rule 926
Rule 3656
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^m}{(c+d x)^{3/2} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {i (a+b x)^m}{2 (i-x) (c+d x)^{3/2}}+\frac {i (a+b x)^m}{2 (i+x) (c+d x)^{3/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i \text {Subst}\left (\int \frac {(a+b x)^m}{(i-x) (c+d x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {i \text {Subst}\left (\int \frac {(a+b x)^m}{(i+x) (c+d x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {\left (i b \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}\right ) \text {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 )^{3/2}} \, dx,x,\tan (e+f x)\right )}{2 (b c-a d) f \sqrt {c+d \tan (e+f x)}}+\frac {\left (i b \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}\right ) \text {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 )^{3/2}} \, dx,x,\tan (e+f x)\right )}{2 (b c-a d) f \sqrt {c+d \tan (e+f x)}}\\ &=\frac {b F_1\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 {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a+b) (b c-a d) f (1+m) \sqrt {c+d \tan (e+f x)}}-\frac {b F_1\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 {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a-b) (b c-a d) f (1+m) \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [F]
time = 9.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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