Optimal. Leaf size=173 \[ \frac {(A+i B) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d}+\frac {(A-i B) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d} \]
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Rubi [A] time = 0.36, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3603, 3602, 130, 511, 510} \[ \frac {(A+i B) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d}+\frac {(A-i B) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 3602
Rule 3603
Rubi steps
\begin {align*} \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=\frac {1}{2} (A-i B) \int (1+i \tan (c+d x)) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \, dx+\frac {1}{2} (A+i B) \int (1-i \tan (c+d x)) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \, dx\\ &=\frac {(A-i B) \operatorname {Subst}\left (\int \frac {\sqrt {x} (a+b x)^n}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {(A+i B) \operatorname {Subst}\left (\int \frac {\sqrt {x} (a+b x)^n}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {(A-i B) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^n}{1-i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {(A+i B) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^n}{1+i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {\left ((A-i B) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a}\right )^n}{1-i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left ((A+i B) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+\frac {b x^2}{a}\right )^n}{1+i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {(A+i B) F_1\left (\frac {3}{2};1,-n;\frac {5}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d}+\frac {(A-i B) F_1\left (\frac {3}{2};1,-n;\frac {5}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d}\\ \end {align*}
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Mathematica [F] time = 1.35, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )}, x\right ) \]
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 = 1.37, size = 0, normalized size = 0.00 \[ \int \left (\sqrt {\tan }\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sqrt {\tan \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n} \sqrt {\tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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