3.1605 \(\int \frac{1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=66 \[ \frac{9 d \sqrt [3]{c+d x}}{4 \sqrt [3]{a+b x} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{4 (a+b x)^{4/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(1/3))/(4*(b*c - a*d)*(a + b*x)^(4/3)) + (9*d*(c + d*x)^(1/3))/(4*(b*c - a*d)^2*(a + b*x)^(1/3))

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Rubi [A]  time = 0.0095416, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{9 d \sqrt [3]{c+d x}}{4 \sqrt [3]{a+b x} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{4 (a+b x)^{4/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/3)*(c + d*x)^(2/3)),x]

[Out]

(-3*(c + d*x)^(1/3))/(4*(b*c - a*d)*(a + b*x)^(4/3)) + (9*d*(c + d*x)^(1/3))/(4*(b*c - a*d)^2*(a + b*x)^(1/3))

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^{7/3} (c+d x)^{2/3}} \, dx &=-\frac{3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}-\frac{(3 d) \int \frac{1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx}{4 (b c-a d)}\\ &=-\frac{3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}+\frac{9 d \sqrt [3]{c+d x}}{4 (b c-a d)^2 \sqrt [3]{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0165527, size = 46, normalized size = 0.7 \[ \frac{3 \sqrt [3]{c+d x} (4 a d-b c+3 b d x)}{4 (a+b x)^{4/3} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/3)*(c + d*x)^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.003, size = 54, normalized size = 0.8 \begin{align*}{\frac{9\,bdx+12\,ad-3\,bc}{4\,{a}^{2}{d}^{2}-8\,abcd+4\,{b}^{2}{c}^{2}}\sqrt [3]{dx+c} \left ( bx+a \right ) ^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/3)*(d*x + c)^(2/3)), x)

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Fricas [B]  time = 2.36251, size = 255, normalized size = 3.86 \begin{align*} \frac{3 \,{\left (3 \, b d x - b c + 4 \, a d\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{4 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(7/3)/(d*x+c)**(2/3),x)

[Out]

Integral(1/((a + b*x)**(7/3)*(c + d*x)**(2/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/3)*(d*x + c)^(2/3)), x)