Optimal. Leaf size=159 \[ \frac {\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {5}{2};1,-n;\frac {7}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{5 d}+\frac {\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {5}{2};1,-n;\frac {7}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{5 d} \]
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Rubi [A] time = 0.22, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3575, 912, 130, 511, 510} \[ \frac {\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {5}{2};1,-n;\frac {7}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{5 d}+\frac {\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {5}{2};1,-n;\frac {7}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 912
Rule 3575
Rubi steps
\begin {align*} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^{3/2} (a+b x)^n}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {i x^{3/2} (a+b x)^n}{2 (i-x)}+\frac {i x^{3/2} (a+b x)^n}{2 (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {x^{3/2} (a+b x)^n}{i-x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \operatorname {Subst}\left (\int \frac {x^{3/2} (a+b x)^n}{i+x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^n}{i-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {i \operatorname {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^n}{i+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {\left (i (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^4 \left (1+\frac {b x^2}{a}\right )^n}{i-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i (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^4 \left (1+\frac {b x^2}{a}\right )^n}{i+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {F_1\left (\frac {5}{2};1,-n;\frac {7}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d}+\frac {F_1\left (\frac {5}{2};1,-n;\frac {7}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{5 d}\\ \end {align*}
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Mathematica [F] time = 0.95, size = 0, normalized size = 0.00 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac {3}{2}}, x\right ) \]
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 = 1.05, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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