3.8.44 \(\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx\) [744]

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {4}{3}}(c+d x)},x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(4/3),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 \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3),x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3), x]

Rubi steps

\begin {align*} \int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx &=\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx\\ \end {align*}

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Mathematica [A]
time = 116.79, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3),x]

[Out]

Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3), x]

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Maple [A]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}{\sec \left (d x +c \right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(4/3),x)

[Out]

int((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(4/3),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((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(4/3),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(5/2)/sec(d*x + c)^(4/3), 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((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(4/3),x, algorithm="fricas")

[Out]

integral((b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2)*sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(4/3), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**(5/2)/sec(d*x+c)**(4/3),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(4/3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(co

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x))^(5/2)/(1/cos(c + d*x))^(4/3),x)

[Out]

int((a + b/cos(c + d*x))^(5/2)/(1/cos(c + d*x))^(4/3), x)

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