3.16.53 \(\int \frac {1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx\)

Optimal. Leaf size=134 \[ -\frac {7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac {1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac {108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac {6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \]

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Rubi [A]  time = 0.03, antiderivative size = 134, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac {1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac {108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac {6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/6)*(c + d*x)^(23/6)),x]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(17/6)) - (108*d*(a + b*x)^(5/6))/(17*(b*c - a*d)^2*(c + d*x)^(17/6)
) - (1296*b*d*(a + b*x)^(5/6))/(187*(b*c - a*d)^3*(c + d*x)^(11/6)) - (7776*b^2*d*(a + b*x)^(5/6))/(935*(b*c -
a*d)^4*(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}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx &=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {(18 d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx}{b c-a d}\\ &=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {(216 b d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{17 (b c-a d)^2}\\ &=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac {\left (1296 b^2 d\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{187 (b c-a d)^3}\\ &=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac {108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac {1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac {7776 b^2 d (a+b x)^{5/6}}{935 (b c-a d)^4 (c+d x)^{5/6}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 118, normalized size = 0.88 \begin {gather*} -\frac {6 \left (55 a^3 d^3-15 a^2 b d^2 (17 c+6 d x)+3 a b^2 d \left (187 c^2+204 c d x+72 d^2 x^2\right )+b^3 \left (935 c^3+3366 c^2 d x+3672 c d^2 x^2+1296 d^3 x^3\right )\right )}{935 \sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(23/6)),x]

[Out]

(-6*(55*a^3*d^3 - 15*a^2*b*d^2*(17*c + 6*d*x) + 3*a*b^2*d*(187*c^2 + 204*c*d*x + 72*d^2*x^2) + b^3*(935*c^3 +
3366*c^2*d*x + 3672*c*d^2*x^2 + 1296*d^3*x^3)))/(935*(b*c - a*d)^4*(a + b*x)^(1/6)*(c + d*x)^(17/6))

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IntegrateAlgebraic [A]  time = 0.14, size = 95, normalized size = 0.71 \begin {gather*} -\frac {6 (a+b x)^{17/6} \left (\frac {935 b^3 (c+d x)^3}{(a+b x)^3}+\frac {561 b^2 d (c+d x)^2}{(a+b x)^2}-\frac {255 b d^2 (c+d x)}{a+b x}+55 d^3\right )}{935 (c+d x)^{17/6} (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(7/6)*(c + d*x)^(23/6)),x]

[Out]

(-6*(a + b*x)^(17/6)*(55*d^3 - (255*b*d^2*(c + d*x))/(a + b*x) + (561*b^2*d*(c + d*x)^2)/(a + b*x)^2 + (935*b^
3*(c + d*x)^3)/(a + b*x)^3))/(935*(b*c - a*d)^4*(c + d*x)^(17/6))

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fricas [B]  time = 1.49, size = 457, normalized size = 3.41 \begin {gather*} -\frac {6 \, {\left (1296 \, b^{3} d^{3} x^{3} + 935 \, b^{3} c^{3} + 561 \, a b^{2} c^{2} d - 255 \, a^{2} b c d^{2} + 55 \, a^{3} d^{3} + 216 \, {\left (17 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (187 \, b^{3} c^{2} d + 34 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{935 \, {\left (a b^{4} c^{7} - 4 \, a^{2} b^{3} c^{6} d + 6 \, a^{3} b^{2} c^{5} d^{2} - 4 \, a^{4} b c^{4} d^{3} + a^{5} c^{3} d^{4} + {\left (b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{4} + {\left (3 \, b^{5} c^{5} d^{2} - 11 \, a b^{4} c^{4} d^{3} + 14 \, a^{2} b^{3} c^{3} d^{4} - 6 \, a^{3} b^{2} c^{2} d^{5} - a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{3} + 3 \, {\left (b^{5} c^{6} d - 3 \, a b^{4} c^{5} d^{2} + 2 \, a^{2} b^{3} c^{4} d^{3} + 2 \, a^{3} b^{2} c^{3} d^{4} - 3 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{2} + {\left (b^{5} c^{7} - a b^{4} c^{6} d - 6 \, a^{2} b^{3} c^{5} d^{2} + 14 \, a^{3} b^{2} c^{4} d^{3} - 11 \, a^{4} b c^{3} d^{4} + 3 \, a^{5} c^{2} d^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(23/6),x, algorithm="fricas")

[Out]

-6/935*(1296*b^3*d^3*x^3 + 935*b^3*c^3 + 561*a*b^2*c^2*d - 255*a^2*b*c*d^2 + 55*a^3*d^3 + 216*(17*b^3*c*d^2 +
a*b^2*d^3)*x^2 + 18*(187*b^3*c^2*d + 34*a*b^2*c*d^2 - 5*a^2*b*d^3)*x)*(b*x + a)^(5/6)*(d*x + c)^(1/6)/(a*b^4*c
^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^4*d^3 - 4*a*b^4*c^3*d^4 + 6*
a^2*b^3*c^2*d^5 - 4*a^3*b^2*c*d^6 + a^4*b*d^7)*x^4 + (3*b^5*c^5*d^2 - 11*a*b^4*c^4*d^3 + 14*a^2*b^3*c^3*d^4 -
6*a^3*b^2*c^2*d^5 - a^4*b*c*d^6 + a^5*d^7)*x^3 + 3*(b^5*c^6*d - 3*a*b^4*c^5*d^2 + 2*a^2*b^3*c^4*d^3 + 2*a^3*b^
2*c^3*d^4 - 3*a^4*b*c^2*d^5 + a^5*c*d^6)*x^2 + (b^5*c^7 - a*b^4*c^6*d - 6*a^2*b^3*c^5*d^2 + 14*a^3*b^2*c^4*d^3
- 11*a^4*b*c^3*d^4 + 3*a^5*c^2*d^5)*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 {7}{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)^(7/6)/(d*x+c)^(23/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)), x)

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maple [A]  time = 0.01, size = 171, normalized size = 1.28 \begin {gather*} -\frac {6 \left (1296 b^{3} d^{3} x^{3}+216 a \,b^{2} d^{3} x^{2}+3672 b^{3} c \,d^{2} x^{2}-90 a^{2} b \,d^{3} x +612 a \,b^{2} c \,d^{2} x +3366 b^{3} c^{2} d x +55 a^{3} d^{3}-255 a^{2} b c \,d^{2}+561 a \,b^{2} c^{2} d +935 b^{3} c^{3}\right )}{935 \left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {17}{6}} \left (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}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(7/6)/(d*x+c)^(23/6),x)

[Out]

-6/935*(1296*b^3*d^3*x^3+216*a*b^2*d^3*x^2+3672*b^3*c*d^2*x^2-90*a^2*b*d^3*x+612*a*b^2*c*d^2*x+3366*b^3*c^2*d*
x+55*a^3*d^3-255*a^2*b*c*d^2+561*a*b^2*c^2*d+935*b^3*c^3)/(b*x+a)^(1/6)/(d*x+c)^(17/6)/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x + a\right )}^{\frac {7}{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)^(7/6)/(d*x+c)^(23/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)), x)

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mupad [B]  time = 1.15, size = 209, normalized size = 1.56 \begin {gather*} -\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {7776\,b^3\,x^3}{935\,{\left (a\,d-b\,c\right )}^4}+\frac {330\,a^3\,d^3-1530\,a^2\,b\,c\,d^2+3366\,a\,b^2\,c^2\,d+5610\,b^3\,c^3}{935\,d^3\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b\,x\,\left (-5\,a^2\,d^2+34\,a\,b\,c\,d+187\,b^2\,c^2\right )}{935\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {1296\,b^2\,x^2\,\left (a\,d+17\,b\,c\right )}{935\,d\,{\left (a\,d-b\,c\right )}^4}\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)^(7/6)*(c + d*x)^(23/6)),x)

[Out]

-((c + d*x)^(1/6)*((7776*b^3*x^3)/(935*(a*d - b*c)^4) + (330*a^3*d^3 + 5610*b^3*c^3 + 3366*a*b^2*c^2*d - 1530*
a^2*b*c*d^2)/(935*d^3*(a*d - b*c)^4) + (108*b*x*(187*b^2*c^2 - 5*a^2*d^2 + 34*a*b*c*d))/(935*d^2*(a*d - b*c)^4
) + (1296*b^2*x^2*(a*d + 17*b*c))/(935*d*(a*d - b*c)^4)))/(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)**(7/6)/(d*x+c)**(23/6),x)

[Out]

Timed out

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