3.8.28 \(\int \sec ^{\frac {5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx\) [728]

Optimal. Leaf size=28 \[ \text {Int}\left (\sec ^{\frac {5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^(5/3)*(a+b*sec(d*x+c))^(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 \sec ^{\frac {5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2),x]

[Out]

Defer[Int][Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2), x]

Rubi steps

\begin {align*} \int \sec ^{\frac {5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx &=\int \sec ^{\frac {5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx\\ \end {align*}

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Mathematica [A]
time = 99.66, size = 0, normalized size = 0.00 \begin {gather*} \int \sec ^{\frac {5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2),x]

[Out]

Integrate[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2), x]

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Maple [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\sec ^{\frac {5}{3}}\left (d x +c \right )\right ) \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(5/3)*(a+b*sec(d*x+c))^(3/2),x)

[Out]

int(sec(d*x+c)^(5/3)*(a+b*sec(d*x+c))^(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(sec(d*x+c)^(5/3)*(a+b*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(3/2)*sec(d*x + c)^(5/3), 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(sec(d*x+c)^(5/3)*(a+b*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**(5/3)*(a+b*sec(d*x+c))**(3/2),x)

[Out]

Timed out

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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(sec(d*x+c)^(5/3)*(a+b*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^(3/2)*sec(d*x + c)^(5/3), 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 (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/3} \,d x \end {gather*}

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))^(5/3),x)

[Out]

int((a + b/cos(c + d*x))^(3/2)*(1/cos(c + d*x))^(5/3), x)

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