3.18.8 \(\int \frac {1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx\) [1708]

Optimal. Leaf size=30 \[ -\frac {4 \sqrt [4]{c+d x}}{(b c-a d) \sqrt [4]{a+b x}} \]

[Out]

-4*(d*x+c)^(1/4)/(-a*d+b*c)/(b*x+a)^(1/4)

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Rubi [A]
time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} -\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(5/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*(c + d*x)^(1/4))/((b*c - a*d)*(a + b*x)^(1/4))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx &=-\frac {4 \sqrt [4]{c+d x}}{(b c-a d) \sqrt [4]{a+b x}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 30, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{c+d x}}{(b c-a d) \sqrt [4]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(5/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*(c + d*x)^(1/4))/((b*c - a*d)*(a + b*x)^(1/4))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((a + b*x)^(5/4)*(c + d*x)^(3/4)),x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.18, size = 27, normalized size = 0.90

method	result	size
gosper	\(\frac {4 \left (d x +c \right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {1}{4}} \left (a d -b c \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(5/4)/(d*x+c)^(3/4),x,method=_RETURNVERBOSE)

[Out]

4/(b*x+a)^(1/4)*(d*x+c)^(1/4)/(a*d-b*c)

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Maxima [F]
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(1/(b*x+a)^(5/4)/(d*x+c)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.29, size = 42, normalized size = 1.40 \begin {gather*} -\frac {4 \, {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(5/4)/(d*x+c)^(3/4),x, algorithm="fricas")

[Out]

-4*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a*b*c - a^2*d + (b^2*c - a*b*d)*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{4}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(5/4)/(d*x+c)**(3/4),x)

[Out]

Integral(1/((a + b*x)**(5/4)*(c + d*x)**(3/4)), x)

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Giac [F] N/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(1/(b*x+a)^(5/4)/(d*x+c)^(3/4),x)

[Out]

Could not integrate

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Mupad [B]
time = 0.71, size = 26, normalized size = 0.87 \begin {gather*} \frac {4\,{\left (c+d\,x\right )}^{1/4}}{\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(c + d*x)^(1/4))/((a*d - b*c)*(a + b*x)^(1/4))

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