3.1836 \(\int \frac{1}{(a+b x)^{7/6} (c+d x)^{17/6}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{432 b d (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^3}-\frac{72 d (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{11/6} (b c-a d)} \]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(11/6)) - (72*d*(a + b*x)^(5/6))/(11*(b*c - a*d)^2*(c + d*x)^(11/6))
- (432*b*d*(a + b*x)^(5/6))/(55*(b*c - a*d)^3*(c + d*x)^(5/6))

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Rubi [A]  time = 0.0202185, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{432 b d (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^3}-\frac{72 d (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{11/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/6)*(c + d*x)^(17/6)),x]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(11/6)) - (72*d*(a + b*x)^(5/6))/(11*(b*c - a*d)^2*(c + d*x)^(11/6))
- (432*b*d*(a + b*x)^(5/6))/(55*(b*c - a*d)^3*(c + d*x)^(5/6))

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)^{7/6} (c+d x)^{17/6}} \, dx &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac{(12 d) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{b c-a d}\\ &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac{72 d (a+b x)^{5/6}}{11 (b c-a d)^2 (c+d x)^{11/6}}-\frac{(72 b d) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{11 (b c-a d)^2}\\ &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{11/6}}-\frac{72 d (a+b x)^{5/6}}{11 (b c-a d)^2 (c+d x)^{11/6}}-\frac{432 b d (a+b x)^{5/6}}{55 (b c-a d)^3 (c+d x)^{5/6}}\\ \end{align*}

Mathematica [A]  time = 0.0299008, size = 77, normalized size = 0.79 \[ -\frac{6 \left (-5 a^2 d^2+2 a b d (11 c+6 d x)+b^2 \left (55 c^2+132 c d x+72 d^2 x^2\right )\right )}{55 \sqrt [6]{a+b x} (c+d x)^{11/6} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(17/6)),x]

[Out]

(-6*(-5*a^2*d^2 + 2*a*b*d*(11*c + 6*d*x) + b^2*(55*c^2 + 132*c*d*x + 72*d^2*x^2)))/(55*(b*c - a*d)^3*(a + b*x)
^(1/6)*(c + d*x)^(11/6))

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Maple [A]  time = 0.006, size = 105, normalized size = 1.1 \begin{align*} -{\frac{-432\,{b}^{2}{d}^{2}{x}^{2}-72\,ab{d}^{2}x-792\,{b}^{2}cdx+30\,{a}^{2}{d}^{2}-132\,abcd-330\,{b}^{2}{c}^{2}}{55\,{a}^{3}{d}^{3}-165\,{a}^{2}cb{d}^{2}+165\,a{b}^{2}{c}^{2}d-55\,{b}^{3}{c}^{3}}{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{11}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(7/6)/(d*x+c)^(17/6),x)

[Out]

-6/55*(-72*b^2*d^2*x^2-12*a*b*d^2*x-132*b^2*c*d*x+5*a^2*d^2-22*a*b*c*d-55*b^2*c^2)/(b*x+a)^(1/6)/(d*x+c)^(11/6
)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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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}{6}}{\left (d x + c\right )}^{\frac{17}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(17/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(17/6)), x)

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Fricas [B]  time = 1.67105, size = 562, normalized size = 5.73 \begin{align*} -\frac{6 \,{\left (72 \, b^{2} d^{2} x^{2} + 55 \, b^{2} c^{2} + 22 \, a b c d - 5 \, a^{2} d^{2} + 12 \,{\left (11 \, b^{2} c d + a b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{55 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(17/6),x, algorithm="fricas")

[Out]

-6/55*(72*b^2*d^2*x^2 + 55*b^2*c^2 + 22*a*b*c*d - 5*a^2*d^2 + 12*(11*b^2*c*d + a*b*d^2)*x)*(b*x + a)^(5/6)*(d*
x + c)^(1/6)/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3
*a^2*b^2*c*d^4 - a^3*b*d^5)*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*
x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x)

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Sympy [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)**(7/6)/(d*x+c)**(17/6),x)

[Out]

Timed out

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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}{6}}{\left (d x + c\right )}^{\frac{17}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(17/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(17/6)), x)