Optimal. Leaf size=56 \[ \frac {2 \tan (c+d x) \sqrt {b \tan ^p(c+d x)} \, _2F_1\left (1,\frac {p+2}{4};\frac {p+6}{4};-\tan ^2(c+d x)\right )}{d (p+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac {2 \tan (c+d x) \sqrt {b \tan ^p(c+d x)} \, _2F_1\left (1,\frac {p+2}{4};\frac {p+6}{4};-\tan ^2(c+d x)\right )}{d (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \sqrt {b \tan ^p(c+d x)} \, dx &=\left (\tan ^{-\frac {p}{2}}(c+d x) \sqrt {b \tan ^p(c+d x)}\right ) \int \tan ^{\frac {p}{2}}(c+d x) \, dx\\ &=\frac {\left (\tan ^{-\frac {p}{2}}(c+d x) \sqrt {b \tan ^p(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^{p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {2 \, _2F_1\left (1,\frac {2+p}{4};\frac {6+p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x) \sqrt {b \tan ^p(c+d x)}}{d (2+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 56, normalized size = 1.00 \[ \frac {2 \tan (c+d x) \sqrt {b \tan ^p(c+d x)} \, _2F_1\left (1,\frac {p+2}{4};\frac {p+6}{4};-\tan ^2(c+d x)\right )}{d (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan \left (d x + c\right )^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \sqrt {b \left (\tan ^{p}\left (d x +c \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan \left (d x + c\right )^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^p} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan ^{p}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________