3.8.82 \(\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx\) [782]

Optimal. Leaf size=28 \[ \text {Int}\left ((d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 21.79, size = 0, normalized size = 0.00 \begin {gather*} \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \end {gather*}

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 = 0.10, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \sec {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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