3.16.16 \(\int \frac {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx\)

Optimal. Leaf size=66 \[ \frac {36 b (a+b x)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

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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)^{7/6}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 (a+b x)^{7/6}}{13 (c+d x)^{13/6} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(7/6))/(13*(b*c - a*d)*(c + d*x)^(13/6)) + (36*b*(a + b*x)^(7/6))/(91*(b*c - a*d)^2*(c + d*x)^(7/
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 {\sqrt [6]{a+b x}}{(c+d x)^{19/6}} \, dx &=\frac {6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {(6 b) \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{13/6}} \, dx}{13 (b c-a d)}\\ &=\frac {6 (a+b x)^{7/6}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {36 b (a+b x)^{7/6}}{91 (b c-a d)^2 (c+d x)^{7/6}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {6 (a+b x)^{7/6} (-7 a d+13 b c+6 b d x)}{91 (c+d x)^{13/6} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(7/6)*(13*b*c - 7*a*d + 6*b*d*x))/(91*(b*c - a*d)^2*(c + d*x)^(13/6))

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IntegrateAlgebraic [A]  time = 0.15, size = 51, normalized size = 0.77 \begin {gather*} \frac {6 (a+b x)^{7/6} \left (13 b-\frac {7 d (a+b x)}{c+d x}\right )}{91 (c+d x)^{7/6} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(1/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(7/6)*(13*b - (7*d*(a + b*x))/(c + d*x)))/(91*(b*c - a*d)^2*(c + d*x)^(7/6))

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fricas [B]  time = 0.93, size = 175, normalized size = 2.65 \begin {gather*} \frac {6 \, {\left (6 \, b^{2} d x^{2} + 13 \, a b c - 7 \, a^{2} d + {\left (13 \, b^{2} c - a b d\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{91 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(19/6),x, algorithm="fricas")

[Out]

6/91*(6*b^2*d*x^2 + 13*a*b*c - 7*a^2*d + (13*b^2*c - a*b*d)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(19/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(19/6), x)

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maple [A]  time = 0.01, size = 54, normalized size = 0.82 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {7}{6}} \left (-6 b d x +7 a d -13 b c \right )}{91 \left (d x +c \right )^{\frac {13}{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((b*x+a)^(1/6)/(d*x+c)^(19/6),x)

[Out]

-6/91*(b*x+a)^(7/6)*(-6*b*d*x+7*a*d-13*b*c)/(d*x+c)^(13/6)/(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 {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(19/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(19/6), x)

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mupad [B]  time = 0.75, size = 137, normalized size = 2.08 \begin {gather*} \frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {36\,b^2\,x^2\,{\left (a+b\,x\right )}^{1/6}}{91\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {\left (42\,a^2\,d-78\,a\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{91\,d^3\,{\left (a\,d-b\,c\right )}^2}+\frac {x\,\left (78\,b^2\,c-6\,a\,b\,d\right )\,{\left (a+b\,x\right )}^{1/6}}{91\,d^3\,{\left (a\,d-b\,c\right )}^2}\right )}{x^3+\frac {c^3}{d^3}+\frac {3\,c\,x^2}{d}+\frac {3\,c^2\,x}{d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(1/6)/(c + d*x)^(19/6),x)

[Out]

((c + d*x)^(5/6)*((36*b^2*x^2*(a + b*x)^(1/6))/(91*d^2*(a*d - b*c)^2) - ((42*a^2*d - 78*a*b*c)*(a + b*x)^(1/6)
)/(91*d^3*(a*d - b*c)^2) + (x*(78*b^2*c - 6*a*b*d)*(a + b*x)^(1/6))/(91*d^3*(a*d - b*c)^2)))/(x^3 + c^3/d^3 +
(3*c*x^2)/d + (3*c^2*x)/d^2)

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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((b*x+a)**(1/6)/(d*x+c)**(19/6),x)

[Out]

Timed out

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