3.1708 \(\int \frac{1}{(a+b x)^{5/4} (c+d x)^{3/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0030378, antiderivative size = 30, normalized size of antiderivative = 1., 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} \[ -\frac{4 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x} (b c-a d)} \]

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*}

Mathematica [A]  time = 0.0070169, size = 30, normalized size = 1. \[ \frac{4 \sqrt [4]{c+d x}}{\sqrt [4]{a+b x} (a d-b c)} \]

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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Maple [A]  time = 0.004, size = 27, normalized size = 0.9 \begin{align*} 4\,{\frac{\sqrt [4]{dx+c}}{\sqrt [4]{bx+a} \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 = 2.31905, size = 97, normalized size = 3.23 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{5}{4}} \left (c + d x\right )^{\frac{3}{4}}}\, dx \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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="giac")

[Out]

Timed out