Optimal. Leaf size=261 \[ \frac {F_1\left (1+m;-\frac {1}{2},1;2+m;-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt {c+d \tan (e+f x)}}{2 (i a+b) f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {F_1\left (1+m;-\frac {1}{2},1;2+m;-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt {c+d \tan (e+f x)}}{2 (i a-b) f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}} \]
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Rubi [A]
time = 0.19, antiderivative size = 261, 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 {\sqrt {c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} F_1\left (m+1;-\frac {1}{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) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {\sqrt {c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} F_1\left (m+1;-\frac {1}{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) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 141
Rule 142
Rule 926
Rule 3656
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^m \sqrt {c+d x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \left (\frac {i (a+b x)^m \sqrt {c+d x}}{2 (i-x)}+\frac {i (a+b x)^m \sqrt {c+d x}}{2 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i \text {Subst}\left (\int \frac {(a+b x)^m \sqrt {c+d x}}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {i \text {Subst}\left (\int \frac {(a+b x)^m \sqrt {c+d x}}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {\left (i \sqrt {c+d \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^m \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {\left (i \sqrt {c+d \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^m \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}\\ &=\frac {F_1\left (1+m;-\frac {1}{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) f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {F_1\left (1+m;-\frac {1}{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) f (1+m) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}\\ \end {align*}
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Mathematica [F]
time = 0.81, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \tan (e+f x))^m \sqrt {c+d \tan (e+f x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \sqrt {c +d \tan \left (f x +e \right )}\, \left (a +b \tan \left (f x +e \right )\right )^{m}\, dx\]
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 \left (a + b \tan {\left (e + f x \right )}\right )^{m} \sqrt {c + d \tan {\left (e + f x \right )}}\, 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 {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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