3.16.45 \(\int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac {432 b^2 \sqrt [6]{a+b x}}{91 \sqrt [6]{c+d x} (b c-a d)^3}+\frac {72 b \sqrt [6]{a+b x}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{13 (c+d x)^{13/6} (b c-a d)} \]

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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 \sqrt [6]{a+b x}}{91 \sqrt [6]{c+d x} (b c-a d)^3}+\frac {72 b \sqrt [6]{a+b x}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{13 (c+d x)^{13/6} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x]

[Out]

(6*(a + b*x)^(1/6))/(13*(b*c - a*d)*(c + d*x)^(13/6)) + (72*b*(a + b*x)^(1/6))/(91*(b*c - a*d)^2*(c + d*x)^(7/
6)) + (432*b^2*(a + b*x)^(1/6))/(91*(b*c - a*d)^3*(c + d*x)^(1/6))

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}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx &=\frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {(12 b) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx}{13 (b c-a d)}\\ &=\frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {72 b \sqrt [6]{a+b x}}{91 (b c-a d)^2 (c+d x)^{7/6}}+\frac {\left (72 b^2\right ) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{91 (b c-a d)^2}\\ &=\frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {72 b \sqrt [6]{a+b x}}{91 (b c-a d)^2 (c+d x)^{7/6}}+\frac {432 b^2 \sqrt [6]{a+b x}}{91 (b c-a d)^3 \sqrt [6]{c+d x}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 0.76 \begin {gather*} \frac {6 \sqrt [6]{a+b x} \left (7 a^2 d^2-2 a b d (13 c+6 d x)+b^2 \left (91 c^2+156 c d x+72 d^2 x^2\right )\right )}{91 (c+d x)^{13/6} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x]

[Out]

(6*(a + b*x)^(1/6)*(7*a^2*d^2 - 2*a*b*d*(13*c + 6*d*x) + b^2*(91*c^2 + 156*c*d*x + 72*d^2*x^2)))/(91*(b*c - a*
d)^3*(c + d*x)^(13/6))

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IntegrateAlgebraic [A]  time = 0.12, size = 83, normalized size = 0.82 \begin {gather*} \frac {6 \left (\frac {91 b^2 \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {7 d^2 (a+b x)^{13/6}}{(c+d x)^{13/6}}-\frac {26 b d (a+b x)^{7/6}}{(c+d x)^{7/6}}\right )}{91 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x]

[Out]

(6*((7*d^2*(a + b*x)^(13/6))/(c + d*x)^(13/6) - (26*b*d*(a + b*x)^(7/6))/(c + d*x)^(7/6) + (91*b^2*(a + b*x)^(
1/6))/(c + d*x)^(1/6)))/(91*(b*c - a*d)^3)

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fricas [B]  time = 1.24, size = 252, normalized size = 2.50 \begin {gather*} \frac {6 \, {\left (72 \, b^{2} d^{2} x^{2} + 91 \, b^{2} c^{2} - 26 \, a b c d + 7 \, a^{2} d^{2} + 12 \, {\left (13 \, b^{2} c d - a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{91 \, {\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)^(5/6)/(d*x+c)^(19/6),x, algorithm="fricas")

[Out]

6/91*(72*b^2*d^2*x^2 + 91*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2 + 12*(13*b^2*c*d - a*b*d^2)*x)*(b*x + a)^(1/6)*(d*x
+ c)^(5/6)/(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 {5}{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(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(19/6)), 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 {1}{6}} \left (72 b^{2} x^{2} d^{2}-12 a b \,d^{2} x +156 b^{2} c d x +7 a^{2} d^{2}-26 a b c d +91 b^{2} c^{2}\right )}{91 \left (d x +c \right )^{\frac {13}{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)^(5/6)/(d*x+c)^(19/6),x)

[Out]

-6/91*(b*x+a)^(1/6)*(72*b^2*d^2*x^2-12*a*b*d^2*x+156*b^2*c*d*x+7*a^2*d^2-26*a*b*c*d+91*b^2*c^2)/(d*x+c)^(13/6)
/(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 {5}{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(1/(b*x+a)^(5/6)/(d*x+c)^(19/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(19/6)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{19/6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^(5/6)*(c + d*x)^(19/6)),x)

[Out]

int(1/((a + b*x)^(5/6)*(c + d*x)^(19/6)), x)

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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)**(5/6)/(d*x+c)**(19/6),x)

[Out]

Timed out

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