3.16.37 \(\int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx\)

Optimal. Leaf size=66 \[ \frac {36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(5/6))/(11*(b*c - a*d)*(c + d*x)^(11/6)) + (36*b*(a + b*x)^(5/6))/(55*(b*c - a*d)^2*(c + d*x)^(5/
6))

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]

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])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx &=\frac {6 (a+b x)^{5/6}}{11 (b c-a d) (c+d x)^{11/6}}+\frac {(6 b) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{11 (b c-a d)}\\ &=\frac {6 (a+b x)^{5/6}}{11 (b c-a d) (c+d x)^{11/6}}+\frac {36 b (a+b x)^{5/6}}{55 (b c-a d)^2 (c+d x)^{5/6}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {6 (a+b x)^{5/6} (-5 a d+11 b c+6 b d x)}{55 (c+d x)^{11/6} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.11, size = 51, normalized size = 0.77 \begin {gather*} \frac {6 (a+b x)^{11/6} \left (\frac {11 b (c+d x)}{a+b x}-5 d\right )}{55 (c+d x)^{11/6} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.38, size = 118, normalized size = 1.79 \begin {gather*} \frac {6 \, {\left (6 \, b d x + 11 \, b c - 5 \, a d\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{55 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/55*(6*b*d*x + 11*b*c - 5*a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^
2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*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 {1}{6}} {\left (d x + c\right )}^{\frac {17}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (-6 b d x +5 a d -11 b c \right )}{55 \left (d x +c \right )^{\frac {11}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 {1}{6}} {\left (d x + c\right )}^{\frac {17}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.86, size = 127, normalized size = 1.92 \begin {gather*} \frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {x\,\left (66\,c\,b^2+6\,a\,d\,b\right )}{55\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {30\,a^2\,d-66\,a\,b\,c}{55\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {36\,b^2\,x^2}{55\,d\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,{\left (a+b\,x\right )}^{1/6}+\frac {c^2\,{\left (a+b\,x\right )}^{1/6}}{d^2}+\frac {2\,c\,x\,{\left (a+b\,x\right )}^{1/6}}{d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c + d*x)^(1/6)*((x*(66*b^2*c + 6*a*b*d))/(55*d^2*(a*d - b*c)^2) - (30*a^2*d - 66*a*b*c)/(55*d^2*(a*d - b*c)^
2) + (36*b^2*x^2)/(55*d*(a*d - b*c)^2)))/(x^2*(a + b*x)^(1/6) + (c^2*(a + b*x)^(1/6))/d^2 + (2*c*x*(a + b*x)^(
1/6))/d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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