3.349 \(\int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m,x\right ) \]

[Out]

Unintegrable[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]

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Rubi [A]  time = 0.0610669, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]

[Out]

Defer[Int][Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]

Rubi steps

\begin{align*} \int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx &=\int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx\\ \end{align*}

Mathematica [A]  time = 0.700808, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]

[Out]

Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]

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Maple [A]  time = 0.3, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\sec \left ( dx+c \right ) } \left ( e\tan \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x)

[Out]

int((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x, algorithm="maxima")

[Out]

integrate(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \tan{\left (c + d x \right )}\right )^{m} \sqrt{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**(1/2)*(e*tan(d*x+c))**m,x)

[Out]

Integral((e*tan(c + d*x))**m*sqrt(a + b*sec(c + d*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)