3.1783 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{23/6}} \, dx\)

Optimal. Leaf size=66 \[ \frac{36 b (a+b x)^{11/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (36*b*(a + b*x)^(11/6))/(187*(b*c - a*d)^2*(c + d*x)^
(11/6))

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Rubi [A]  time = 0.0090605, 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} \[ \frac{36 b (a+b x)^{11/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/6)/(c + d*x)^(23/6),x]

[Out]

(6*(a + b*x)^(11/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (36*b*(a + b*x)^(11/6))/(187*(b*c - a*d)^2*(c + d*x)^
(11/6))

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{(a+b x)^{5/6}}{(c+d x)^{23/6}} \, dx &=\frac{6 (a+b x)^{11/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac{(6 b) \int \frac{(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx}{17 (b c-a d)}\\ &=\frac{6 (a+b x)^{11/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac{36 b (a+b x)^{11/6}}{187 (b c-a d)^2 (c+d x)^{11/6}}\\ \end{align*}

Mathematica [A]  time = 0.0229957, size = 46, normalized size = 0.7 \[ \frac{6 (a+b x)^{11/6} (-11 a d+17 b c+6 b d x)}{187 (c+d x)^{17/6} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/6)/(c + d*x)^(23/6),x]

[Out]

(6*(a + b*x)^(11/6)*(17*b*c - 11*a*d + 6*b*d*x))/(187*(b*c - a*d)^2*(c + d*x)^(17/6))

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Maple [A]  time = 0.003, size = 54, normalized size = 0.8 \begin{align*} -{\frac{-36\,bdx+66\,ad-102\,bc}{187\,{a}^{2}{d}^{2}-374\,abcd+187\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{{\frac{11}{6}}} \left ( dx+c \right ) ^{-{\frac{17}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/6)/(d*x+c)^(23/6),x)

[Out]

-6/187*(b*x+a)^(11/6)*(-6*b*d*x+11*a*d-17*b*c)/(d*x+c)^(17/6)/(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{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(23/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(23/6), x)

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Fricas [B]  time = 1.83813, size = 378, normalized size = 5.73 \begin{align*} \frac{6 \,{\left (6 \, b^{2} d x^{2} + 17 \, a b c - 11 \, a^{2} d +{\left (17 \, b^{2} c - 5 \, a b d\right )} x\right )}{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{187 \,{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{3} + 3 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2} + 3 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(23/6),x, algorithm="fricas")

[Out]

6/187*(6*b^2*d*x^2 + 17*a*b*c - 11*a^2*d + (17*b^2*c - 5*a*b*d)*x)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(b^2*c^5 -
2*a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^3 + 3*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c
*d^4)*x^2 + 3*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*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((b*x+a)**(5/6)/(d*x+c)**(23/6),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/6)/(d*x+c)^(23/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(23/6), x)