3.736 \(\int \frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx\)

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{3}}(c+d x)},x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/3),x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3),x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3), x]

Rubi steps

\begin {align*} \int \frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx &=\int \frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx\\ \end {align*}

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Mathematica [A]  time = 39.22, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3),x]

[Out]

Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3), x]

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fricas [A]  time = 2.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {7}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/3),x, algorithm="fricas")

[Out]

integral((b*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(7/3), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{\sec \left (d x +c \right )^{\frac {7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/3),x)

[Out]

int((a+b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/3),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(3/2)/sec(d*x+c)^(7/3),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(7/3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x))^(3/2)/(1/cos(c + d*x))^(7/3),x)

[Out]

int((a + b/cos(c + d*x))^(3/2)/(1/cos(c + d*x))^(7/3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**(3/2)/sec(d*x+c)**(7/3),x)

[Out]

Timed out

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