3.15.93 \(\int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=32 \[ -\frac {4 (c+d x)^{3/4}}{3 (a+b x)^{3/4} (b c-a d)} \]

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Rubi [A]  time = 0.00, antiderivative size = 32, 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 (c+d x)^{3/4}}{3 (a+b x)^{3/4} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x]

[Out]

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

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)^{7/4} \sqrt [4]{c+d x}} \, dx &=-\frac {4 (c+d x)^{3/4}}{3 (b c-a d) (a+b x)^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} -\frac {4 (c+d x)^{3/4}}{3 (a+b x)^{3/4} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 32, normalized size = 1.00 \begin {gather*} -\frac {4 (c+d x)^{3/4}}{3 (a+b x)^{3/4} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x]

[Out]

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

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fricas [A]  time = 1.10, size = 42, normalized size = 1.31 \begin {gather*} -\frac {4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{3 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

-4/3*(b*x + a)^(1/4)*(d*x + c)^(3/4)/(a*b*c - a^2*d + (b^2*c - a*b*d)*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 {7}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/4)*(d*x + c)^(1/4)), x)

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maple [A]  time = 0.00, size = 27, normalized size = 0.84 \begin {gather*} \frac {4 \left (d x +c \right )^{\frac {3}{4}}}{3 \left (b x +a \right )^{\frac {3}{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)^(7/4)/(d*x+c)^(1/4),x)

[Out]

4/3/(b*x+a)^(3/4)*(d*x+c)^(3/4)/(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 {7}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/4)*(d*x + c)^(1/4)), 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 )}^{7/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(7/4)*(c + d*x)^(1/4)),x)

[Out]

int(1/((a + b*x)^(7/4)*(c + d*x)^(1/4)), 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 {7}{4}} \sqrt [4]{c + d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(7/4)/(d*x+c)**(1/4),x)

[Out]

Integral(1/((a + b*x)**(7/4)*(c + d*x)**(1/4)), x)

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