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

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

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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]{a+b x}}{\sqrt [4]{c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 30, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 42, normalized size = 1.40 \begin {gather*} \frac {4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 27, normalized size = 0.90 \begin {gather*} -\frac {4 \left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}} \left (a d -b c \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/((a + b*x)^(3/4)*(c + d*x)^(5/4)), 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 {3}{4}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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