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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \]

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

________________________________________________________________________________________

Mathematica [A]  time = 2.74, size = 0, normalized size = 0.00 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}}{b^{2} \sec \left (f x + e\right )^{2} + 2 \, a b \sec \left (f x + e\right ) + a^{2}}, x\right ) \]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

maple [A]  time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x +e \right )\right )^{n}}{\left (a +b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________