3.1785 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx\)

Optimal. Leaf size=136 \[ \frac{7776 b^3 (a+b x)^{11/6}}{124729 (c+d x)^{11/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{11/6}}{11339 (c+d x)^{17/6} (b c-a d)^3}+\frac{108 b (a+b x)^{11/6}}{667 (c+d x)^{23/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6))/(29*(b*c - a*d)*(c + d*x)^(29/6)) + (108*b*(a + b*x)^(11/6))/(667*(b*c - a*d)^2*(c + d*x)
^(23/6)) + (1296*b^2*(a + b*x)^(11/6))/(11339*(b*c - a*d)^3*(c + d*x)^(17/6)) + (7776*b^3*(a + b*x)^(11/6))/(1
24729*(b*c - a*d)^4*(c + d*x)^(11/6))

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Rubi [A]  time = 0.0334923, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{7776 b^3 (a+b x)^{11/6}}{124729 (c+d x)^{11/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{11/6}}{11339 (c+d x)^{17/6} (b c-a d)^3}+\frac{108 b (a+b x)^{11/6}}{667 (c+d x)^{23/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/6)/(c + d*x)^(35/6),x]

[Out]

(6*(a + b*x)^(11/6))/(29*(b*c - a*d)*(c + d*x)^(29/6)) + (108*b*(a + b*x)^(11/6))/(667*(b*c - a*d)^2*(c + d*x)
^(23/6)) + (1296*b^2*(a + b*x)^(11/6))/(11339*(b*c - a*d)^3*(c + d*x)^(17/6)) + (7776*b^3*(a + b*x)^(11/6))/(1
24729*(b*c - a*d)^4*(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)^{35/6}} \, dx &=\frac{6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac{(18 b) \int \frac{(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx}{29 (b c-a d)}\\ &=\frac{6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac{108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac{\left (216 b^2\right ) \int \frac{(a+b x)^{5/6}}{(c+d x)^{23/6}} \, dx}{667 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac{108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac{1296 b^2 (a+b x)^{11/6}}{11339 (b c-a d)^3 (c+d x)^{17/6}}+\frac{\left (1296 b^3\right ) \int \frac{(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx}{11339 (b c-a d)^3}\\ &=\frac{6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac{108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac{1296 b^2 (a+b x)^{11/6}}{11339 (b c-a d)^3 (c+d x)^{17/6}}+\frac{7776 b^3 (a+b x)^{11/6}}{124729 (b c-a d)^4 (c+d x)^{11/6}}\\ \end{align*}

Mathematica [A]  time = 0.0604629, size = 118, normalized size = 0.87 \[ \frac{6 (a+b x)^{11/6} \left (561 a^2 b d^2 (29 c+6 d x)-4301 a^3 d^3-33 a b^2 d \left (667 c^2+348 c d x+72 d^2 x^2\right )+b^3 \left (12006 c^2 d x+11339 c^3+6264 c d^2 x^2+1296 d^3 x^3\right )\right )}{124729 (c+d x)^{29/6} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/6)/(c + d*x)^(35/6),x]

[Out]

(6*(a + b*x)^(11/6)*(-4301*a^3*d^3 + 561*a^2*b*d^2*(29*c + 6*d*x) - 33*a*b^2*d*(667*c^2 + 348*c*d*x + 72*d^2*x
^2) + b^3*(11339*c^3 + 12006*c^2*d*x + 6264*c*d^2*x^2 + 1296*d^3*x^3)))/(124729*(b*c - a*d)^4*(c + d*x)^(29/6)
)

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Maple [A]  time = 0.008, size = 171, normalized size = 1.3 \begin{align*} -{\frac{-7776\,{x}^{3}{b}^{3}{d}^{3}+14256\,a{b}^{2}{d}^{3}{x}^{2}-37584\,{b}^{3}c{d}^{2}{x}^{2}-20196\,{a}^{2}b{d}^{3}x+68904\,a{b}^{2}c{d}^{2}x-72036\,{b}^{3}{c}^{2}dx+25806\,{a}^{3}{d}^{3}-97614\,{a}^{2}cb{d}^{2}+132066\,a{b}^{2}{c}^{2}d-68034\,{b}^{3}{c}^{3}}{124729\,{a}^{4}{d}^{4}-498916\,{a}^{3}bc{d}^{3}+748374\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-498916\,{b}^{3}d{c}^{3}a+124729\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{{\frac{11}{6}}} \left ( dx+c \right ) ^{-{\frac{29}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/6)/(d*x+c)^(35/6),x)

[Out]

-6/124729*(b*x+a)^(11/6)*(-1296*b^3*d^3*x^3+2376*a*b^2*d^3*x^2-6264*b^3*c*d^2*x^2-3366*a^2*b*d^3*x+11484*a*b^2
*c*d^2*x-12006*b^3*c^2*d*x+4301*a^3*d^3-16269*a^2*b*c*d^2+22011*a*b^2*c^2*d-11339*b^3*c^3)/(d*x+c)^(29/6)/(a^4
*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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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{35}{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)^(35/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(35/6), x)

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Fricas [B]  time = 1.90939, size = 1131, normalized size = 8.32 \begin{align*} \frac{6 \,{\left (1296 \, b^{4} d^{3} x^{4} + 11339 \, a b^{3} c^{3} - 22011 \, a^{2} b^{2} c^{2} d + 16269 \, a^{3} b c d^{2} - 4301 \, a^{4} d^{3} + 216 \,{\left (29 \, b^{4} c d^{2} - 5 \, a b^{3} d^{3}\right )} x^{3} + 18 \,{\left (667 \, b^{4} c^{2} d - 290 \, a b^{3} c d^{2} + 55 \, a^{2} b^{2} d^{3}\right )} x^{2} +{\left (11339 \, b^{4} c^{3} - 10005 \, a b^{3} c^{2} d + 4785 \, a^{2} b^{2} c d^{2} - 935 \, a^{3} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{124729 \,{\left (b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4} +{\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} x^{5} + 5 \,{\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} x^{4} + 10 \,{\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} x^{3} + 10 \,{\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} x^{2} + 5 \,{\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\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)^(35/6),x, algorithm="fricas")

[Out]

6/124729*(1296*b^4*d^3*x^4 + 11339*a*b^3*c^3 - 22011*a^2*b^2*c^2*d + 16269*a^3*b*c*d^2 - 4301*a^4*d^3 + 216*(2
9*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 18*(667*b^4*c^2*d - 290*a*b^3*c*d^2 + 55*a^2*b^2*d^3)*x^2 + (11339*b^4*c^3 -
10005*a*b^3*c^2*d + 4785*a^2*b^2*c*d^2 - 935*a^3*b*d^3)*x)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(b^4*c^9 - 4*a*b^3*
c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4 + (b^4*c^4*d^5 - 4*a*b^3*c^3*d^6 + 6*a^2*b^2*c^2*d^7
- 4*a^3*b*c*d^8 + a^4*d^9)*x^5 + 5*(b^4*c^5*d^4 - 4*a*b^3*c^4*d^5 + 6*a^2*b^2*c^3*d^6 - 4*a^3*b*c^2*d^7 + a^4
*c*d^8)*x^4 + 10*(b^4*c^6*d^3 - 4*a*b^3*c^5*d^4 + 6*a^2*b^2*c^4*d^5 - 4*a^3*b*c^3*d^6 + a^4*c^2*d^7)*x^3 + 10*
(b^4*c^7*d^2 - 4*a*b^3*c^6*d^3 + 6*a^2*b^2*c^5*d^4 - 4*a^3*b*c^4*d^5 + a^4*c^3*d^6)*x^2 + 5*(b^4*c^8*d - 4*a*b
^3*c^7*d^2 + 6*a^2*b^2*c^6*d^3 - 4*a^3*b*c^5*d^4 + a^4*c^4*d^5)*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)**(35/6),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{35}{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)^(35/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(35/6), x)