3.782 \(\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left ((a+b \sec (e+f x))^{3/2} (d \sec (e+f x))^n,x\right ) \]

[Out]

Unintegrable[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]

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Rubi [A]  time = 0.0686343, 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 = {} \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2),x]

[Out]

Defer[Int][(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]

Rubi steps

\begin{align*} \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx &=\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx\\ \end{align*}

Mathematica [A]  time = 13.5194, size = 0, normalized size = 0. \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2),x]

[Out]

Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]

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Maple [A]  time = 0.204, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{n} \left ( a+b\sec \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x)

[Out]

int((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(3/2)*(d*sec(f*x + e))^n, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \left (d \sec \left (f x + e\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e) + a)^(3/2)*(d*sec(f*x + e))^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))**n*(a+b*sec(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(3/2)*(d*sec(f*x + e))^n, x)