3.16.38 \(\int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac {432 b^2 (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^3}+\frac {72 b (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

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Rubi [A]  time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {432 b^2 (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^3}+\frac {72 b (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (72*b*(a + b*x)^(5/6))/(187*(b*c - a*d)^2*(c + d*x)^(1
1/6)) + (432*b^2*(a + b*x)^(5/6))/(935*(b*c - a*d)^3*(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)^{23/6}} \, dx &=\frac {6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac {(12 b) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{17 (b c-a d)}\\ &=\frac {6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac {72 b (a+b x)^{5/6}}{187 (b c-a d)^2 (c+d x)^{11/6}}+\frac {\left (72 b^2\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{187 (b c-a d)^2}\\ &=\frac {6 (a+b x)^{5/6}}{17 (b c-a d) (c+d x)^{17/6}}+\frac {72 b (a+b x)^{5/6}}{187 (b c-a d)^2 (c+d x)^{11/6}}+\frac {432 b^2 (a+b x)^{5/6}}{935 (b c-a d)^3 (c+d x)^{5/6}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 77, normalized size = 0.76 \begin {gather*} \frac {6 (a+b x)^{5/6} \left (55 a^2 d^2-10 a b d (17 c+6 d x)+b^2 \left (187 c^2+204 c d x+72 d^2 x^2\right )\right )}{935 (c+d x)^{17/6} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6)*(55*a^2*d^2 - 10*a*b*d*(17*c + 6*d*x) + b^2*(187*c^2 + 204*c*d*x + 72*d^2*x^2)))/(935*(b*c
- a*d)^3*(c + d*x)^(17/6))

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IntegrateAlgebraic [A]  time = 0.12, size = 73, normalized size = 0.72 \begin {gather*} \frac {6 (a+b x)^{17/6} \left (\frac {187 b^2 (c+d x)^2}{(a+b x)^2}-\frac {170 b d (c+d x)}{a+b x}+55 d^2\right )}{935 (c+d x)^{17/6} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(17/6)*(55*d^2 - (170*b*d*(c + d*x))/(a + b*x) + (187*b^2*(c + d*x)^2)/(a + b*x)^2))/(935*(b*c -
a*d)^3*(c + d*x)^(17/6))

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fricas [B]  time = 1.37, size = 252, normalized size = 2.50 \begin {gather*} \frac {6 \, {\left (72 \, b^{2} d^{2} x^{2} + 187 \, b^{2} c^{2} - 170 \, a b c d + 55 \, a^{2} d^{2} + 12 \, {\left (17 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{935 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{3} + 3 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} + 3 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\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)^(23/6),x, algorithm="fricas")

[Out]

6/935*(72*b^2*d^2*x^2 + 187*b^2*c^2 - 170*a*b*c*d + 55*a^2*d^2 + 12*(17*b^2*c*d - 5*a*b*d^2)*x)*(b*x + a)^(5/6
)*(d*x + c)^(1/6)/(b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 +
3*a^2*b*c*d^5 - a^3*d^6)*x^3 + 3*(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^2 + 3*(b^3*c^
5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*d^4)*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 {23}{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)^(23/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)

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maple [A]  time = 0.01, size = 105, normalized size = 1.04 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (72 b^{2} x^{2} d^{2}-60 a b \,d^{2} x +204 b^{2} c d x +55 a^{2} d^{2}-170 a b c d +187 b^{2} c^{2}\right )}{935 \left (d x +c \right )^{\frac {17}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x)

[Out]

-6/935*(b*x+a)^(5/6)*(72*b^2*d^2*x^2-60*a*b*d^2*x+204*b^2*c*d*x+55*a^2*d^2-170*a*b*c*d+187*b^2*c^2)/(d*x+c)^(1
7/6)/(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 = 0.00, 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 {23}{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)^(23/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)

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mupad [B]  time = 1.03, size = 203, normalized size = 2.01 \begin {gather*} -\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {330\,a^3\,d^2-1020\,a^2\,b\,c\,d+1122\,a\,b^2\,c^2}{935\,d^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x\,\left (-30\,a^2\,b\,d^2+204\,a\,b^2\,c\,d+1122\,b^3\,c^2\right )}{935\,d^3\,{\left (a\,d-b\,c\right )}^3}+\frac {432\,b^3\,x^3}{935\,d\,{\left (a\,d-b\,c\right )}^3}+\frac {72\,b^2\,x^2\,\left (a\,d+17\,b\,c\right )}{935\,d^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^3\,{\left (a+b\,x\right )}^{1/6}+\frac {c^3\,{\left (a+b\,x\right )}^{1/6}}{d^3}+\frac {3\,c\,x^2\,{\left (a+b\,x\right )}^{1/6}}{d}+\frac {3\,c^2\,x\,{\left (a+b\,x\right )}^{1/6}}{d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x)

[Out]

-((c + d*x)^(1/6)*((330*a^3*d^2 + 1122*a*b^2*c^2 - 1020*a^2*b*c*d)/(935*d^3*(a*d - b*c)^3) + (x*(1122*b^3*c^2
- 30*a^2*b*d^2 + 204*a*b^2*c*d))/(935*d^3*(a*d - b*c)^3) + (432*b^3*x^3)/(935*d*(a*d - b*c)^3) + (72*b^2*x^2*(
a*d + 17*b*c))/(935*d^2*(a*d - b*c)^3)))/(x^3*(a + b*x)^(1/6) + (c^3*(a + b*x)^(1/6))/d^3 + (3*c*x^2*(a + b*x)
^(1/6))/d + (3*c^2*x*(a + b*x)^(1/6))/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(1/(b*x+a)**(1/6)/(d*x+c)**(23/6),x)

[Out]

Timed out

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