Optimal. Leaf size=187 \[ \frac {(A+i B) \tan ^{m+1}(c+d x) \sqrt {a+b \tan (c+d x)} F_1\left (m+1;-\frac {1}{2},1;m+2;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (m+1) \sqrt {\frac {b \tan (c+d x)}{a}+1}}+\frac {(A-i B) \tan ^{m+1}(c+d x) \sqrt {a+b \tan (c+d x)} F_1\left (m+1;-\frac {1}{2},1;m+2;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (m+1) \sqrt {\frac {b \tan (c+d x)}{a}+1}} \]
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Rubi [A] time = 0.40, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3603, 3602, 135, 133} \[ \frac {(A+i B) \tan ^{m+1}(c+d x) \sqrt {a+b \tan (c+d x)} F_1\left (m+1;-\frac {1}{2},1;m+2;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (m+1) \sqrt {\frac {b \tan (c+d x)}{a}+1}}+\frac {(A-i B) \tan ^{m+1}(c+d x) \sqrt {a+b \tan (c+d x)} F_1\left (m+1;-\frac {1}{2},1;m+2;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (m+1) \sqrt {\frac {b \tan (c+d x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 3602
Rule 3603
Rubi steps
\begin {align*} \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx &=\frac {1}{2} (A-i B) \int (1+i \tan (c+d x)) \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} \, dx+\frac {1}{2} (A+i B) \int (1-i \tan (c+d x)) \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\\ &=\frac {(A-i B) \operatorname {Subst}\left (\int \frac {x^m \sqrt {a+b x}}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {(A+i B) \operatorname {Subst}\left (\int \frac {x^m \sqrt {a+b x}}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {\left ((A-i B) \sqrt {a+b \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^m \sqrt {1+\frac {b x}{a}}}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d \sqrt {1+\frac {b \tan (c+d x)}{a}}}+\frac {\left ((A+i B) \sqrt {a+b \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^m \sqrt {1+\frac {b x}{a}}}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d \sqrt {1+\frac {b \tan (c+d x)}{a}}}\\ &=\frac {(A+i B) F_1\left (1+m;-\frac {1}{2},1;2+m;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d (1+m) \sqrt {1+\frac {b \tan (c+d x)}{a}}}+\frac {(A-i B) F_1\left (1+m;-\frac {1}{2},1;2+m;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d (1+m) \sqrt {1+\frac {b \tan (c+d x)}{a}}}\\ \end {align*}
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Mathematica [F] time = 5.29, size = 0, normalized size = 0.00 \[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{m}, x\right ) \]
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] time = 1.48, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{m}\left (d x +c \right )\right ) \sqrt {a +b \tan \left (d x +c \right )}\, \left (A +B \tan \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]
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 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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