Optimal. Leaf size=170 \[ \frac {2 (A-C) \sqrt {b \tan (c+d x)} \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {1}{4} (2 m+3);\frac {1}{4} (2 m+7);-\tan ^2(c+d x)\right )}{d (2 m+3)}+\frac {2 B \sqrt {b \tan (c+d x)} \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {1}{4} (2 m+5);\frac {1}{4} (2 m+9);-\tan ^2(c+d x)\right )}{d (2 m+5)}+\frac {2 C \sqrt {b \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)} \]
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Rubi [A] time = 0.14, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {20, 3630, 3538, 3476, 364} \[ \frac {2 (A-C) \sqrt {b \tan (c+d x)} \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {1}{4} (2 m+3);\frac {1}{4} (2 m+7);-\tan ^2(c+d x)\right )}{d (2 m+3)}+\frac {2 B \sqrt {b \tan (c+d x)} \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {1}{4} (2 m+5);\frac {1}{4} (2 m+9);-\tan ^2(c+d x)\right )}{d (2 m+5)}+\frac {2 C \sqrt {b \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 3476
Rule 3538
Rule 3630
Rubi steps
\begin {align*} \int \tan ^m(c+d x) \sqrt {b \tan (c+d x)} \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\frac {\sqrt {b \tan (c+d x)} \int \tan ^{\frac {1}{2}+m}(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx}{\sqrt {\tan (c+d x)}}\\ &=\frac {2 C \tan ^{1+m}(c+d x) \sqrt {b \tan (c+d x)}}{d (3+2 m)}+\frac {\sqrt {b \tan (c+d x)} \int \tan ^{\frac {1}{2}+m}(c+d x) (A-C+B \tan (c+d x)) \, dx}{\sqrt {\tan (c+d x)}}\\ &=\frac {2 C \tan ^{1+m}(c+d x) \sqrt {b \tan (c+d x)}}{d (3+2 m)}+\frac {\left (B \sqrt {b \tan (c+d x)}\right ) \int \tan ^{\frac {3}{2}+m}(c+d x) \, dx}{\sqrt {\tan (c+d x)}}+\frac {\left ((A-C) \sqrt {b \tan (c+d x)}\right ) \int \tan ^{\frac {1}{2}+m}(c+d x) \, dx}{\sqrt {\tan (c+d x)}}\\ &=\frac {2 C \tan ^{1+m}(c+d x) \sqrt {b \tan (c+d x)}}{d (3+2 m)}+\frac {\left (B \sqrt {b \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^{\frac {3}{2}+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}+\frac {\left ((A-C) \sqrt {b \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^{\frac {1}{2}+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt {\tan (c+d x)}}\\ &=\frac {2 C \tan ^{1+m}(c+d x) \sqrt {b \tan (c+d x)}}{d (3+2 m)}+\frac {2 (A-C) \, _2F_1\left (1,\frac {1}{4} (3+2 m);\frac {1}{4} (7+2 m);-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {b \tan (c+d x)}}{d (3+2 m)}+\frac {2 B \, _2F_1\left (1,\frac {1}{4} (5+2 m);\frac {1}{4} (9+2 m);-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x) \sqrt {b \tan (c+d x)}}{d (5+2 m)}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 133, normalized size = 0.78 \[ \frac {2 \sqrt {b \tan (c+d x)} \tan ^{m+1}(c+d x) \left ((2 m+5) (A-C) \, _2F_1\left (1,\frac {1}{4} (2 m+3);\frac {1}{4} (2 m+7);-\tan ^2(c+d x)\right )+B (2 m+3) \tan (c+d x) \, _2F_1\left (1,\frac {1}{4} (2 m+5);\frac {1}{4} (2 m+9);-\tan ^2(c+d x)\right )+C (2 m+5)\right )}{d (2 m+3) (2 m+5)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right )} \tan \left (d x + c\right )^{m}, x\right ) \]
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 = 1.71, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{m}\left (d x +c \right )\right ) \sqrt {b \tan \left (d x +c \right )}\, \left (A +B \tan \left (d x +c \right )+C \left (\tan ^{2}\left (d x +c \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^m\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}\,\left (C\,{\mathrm {tan}\left (c+d\,x\right )}^2+B\,\mathrm {tan}\left (c+d\,x\right )+A\right ) \,d x \]
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 \[ \int \sqrt {b \tan {\left (c + d x \right )}} \left (A + B \tan {\left (c + d x \right )} + C \tan ^{2}{\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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