3.1804 \(\int \frac{(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx\)

Optimal. Leaf size=136 \[ \frac{7776 b^3 (a+b x)^{13/6}}{191425 (c+d x)^{13/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{13/6}}{14725 (c+d x)^{19/6} (b c-a d)^3}+\frac{108 b (a+b x)^{13/6}}{775 (c+d x)^{25/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(13/6))/(31*(b*c - a*d)*(c + d*x)^(31/6)) + (108*b*(a + b*x)^(13/6))/(775*(b*c - a*d)^2*(c + d*x)
^(25/6)) + (1296*b^2*(a + b*x)^(13/6))/(14725*(b*c - a*d)^3*(c + d*x)^(19/6)) + (7776*b^3*(a + b*x)^(13/6))/(1
91425*(b*c - a*d)^4*(c + d*x)^(13/6))

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Rubi [A]  time = 0.0296088, antiderivative size = 136, normalized size of antiderivative = 1., 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} \[ \frac{7776 b^3 (a+b x)^{13/6}}{191425 (c+d x)^{13/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{13/6}}{14725 (c+d x)^{19/6} (b c-a d)^3}+\frac{108 b (a+b x)^{13/6}}{775 (c+d x)^{25/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/6)/(c + d*x)^(37/6),x]

[Out]

(6*(a + b*x)^(13/6))/(31*(b*c - a*d)*(c + d*x)^(31/6)) + (108*b*(a + b*x)^(13/6))/(775*(b*c - a*d)^2*(c + d*x)
^(25/6)) + (1296*b^2*(a + b*x)^(13/6))/(14725*(b*c - a*d)^3*(c + d*x)^(19/6)) + (7776*b^3*(a + b*x)^(13/6))/(1
91425*(b*c - a*d)^4*(c + d*x)^(13/6))

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])

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]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx &=\frac{6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac{(18 b) \int \frac{(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx}{31 (b c-a d)}\\ &=\frac{6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac{108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac{\left (216 b^2\right ) \int \frac{(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx}{775 (b c-a d)^2}\\ &=\frac{6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac{108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac{1296 b^2 (a+b x)^{13/6}}{14725 (b c-a d)^3 (c+d x)^{19/6}}+\frac{\left (1296 b^3\right ) \int \frac{(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{14725 (b c-a d)^3}\\ &=\frac{6 (a+b x)^{13/6}}{31 (b c-a d) (c+d x)^{31/6}}+\frac{108 b (a+b x)^{13/6}}{775 (b c-a d)^2 (c+d x)^{25/6}}+\frac{1296 b^2 (a+b x)^{13/6}}{14725 (b c-a d)^3 (c+d x)^{19/6}}+\frac{7776 b^3 (a+b x)^{13/6}}{191425 (b c-a d)^4 (c+d x)^{13/6}}\\ \end{align*}

Mathematica [A]  time = 0.0672193, size = 118, normalized size = 0.87 \[ \frac{6 (a+b x)^{13/6} \left (741 a^2 b d^2 (31 c+6 d x)-6175 a^3 d^3-39 a b^2 d \left (775 c^2+372 c d x+72 d^2 x^2\right )+b^3 \left (13950 c^2 d x+14725 c^3+6696 c d^2 x^2+1296 d^3 x^3\right )\right )}{191425 (c+d x)^{31/6} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(7/6)/(c + d*x)^(37/6),x]

[Out]

(6*(a + b*x)^(13/6)*(-6175*a^3*d^3 + 741*a^2*b*d^2*(31*c + 6*d*x) - 39*a*b^2*d*(775*c^2 + 372*c*d*x + 72*d^2*x
^2) + b^3*(14725*c^3 + 13950*c^2*d*x + 6696*c*d^2*x^2 + 1296*d^3*x^3)))/(191425*(b*c - a*d)^4*(c + d*x)^(31/6)
)

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Maple [A]  time = 0.008, size = 171, normalized size = 1.3 \begin{align*} -{\frac{-7776\,{x}^{3}{b}^{3}{d}^{3}+16848\,a{b}^{2}{d}^{3}{x}^{2}-40176\,{b}^{3}c{d}^{2}{x}^{2}-26676\,{a}^{2}b{d}^{3}x+87048\,a{b}^{2}c{d}^{2}x-83700\,{b}^{3}{c}^{2}dx+37050\,{a}^{3}{d}^{3}-137826\,{a}^{2}cb{d}^{2}+181350\,a{b}^{2}{c}^{2}d-88350\,{b}^{3}{c}^{3}}{191425\,{a}^{4}{d}^{4}-765700\,{a}^{3}bc{d}^{3}+1148550\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-765700\,a{b}^{3}{c}^{3}d+191425\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{{\frac{13}{6}}} \left ( dx+c \right ) ^{-{\frac{31}{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(7/6)/(d*x+c)^(37/6),x)

[Out]

-6/191425*(b*x+a)^(13/6)*(-1296*b^3*d^3*x^3+2808*a*b^2*d^3*x^2-6696*b^3*c*d^2*x^2-4446*a^2*b*d^3*x+14508*a*b^2
*c*d^2*x-13950*b^3*c^2*d*x+6175*a^3*d^3-22971*a^2*b*c*d^2+30225*a*b^2*c^2*d-14725*b^3*c^3)/(d*x+c)^(31/6)/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{7}{6}}}{{\left (d x + c\right )}^{\frac{37}{6}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(37/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(37/6), x)

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Fricas [B]  time = 1.88865, size = 1373, normalized size = 10.1 \begin{align*} \frac{6 \,{\left (1296 \, b^{5} d^{3} x^{5} + 14725 \, a^{2} b^{3} c^{3} - 30225 \, a^{3} b^{2} c^{2} d + 22971 \, a^{4} b c d^{2} - 6175 \, a^{5} d^{3} + 216 \,{\left (31 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 18 \,{\left (775 \, b^{5} c^{2} d - 62 \, a b^{4} c d^{2} + 7 \, a^{2} b^{3} d^{3}\right )} x^{3} +{\left (14725 \, b^{5} c^{3} - 2325 \, a b^{4} c^{2} d + 651 \, a^{2} b^{3} c d^{2} - 91 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (14725 \, a b^{4} c^{3} - 23250 \, a^{2} b^{3} c^{2} d + 15717 \, a^{3} b^{2} c d^{2} - 3952 \, a^{4} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{191425 \,{\left (b^{4} c^{10} - 4 \, a b^{3} c^{9} d + 6 \, a^{2} b^{2} c^{8} d^{2} - 4 \, a^{3} b c^{7} d^{3} + a^{4} c^{6} d^{4} +{\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{6} + 6 \,{\left (b^{4} c^{5} d^{5} - 4 \, a b^{3} c^{4} d^{6} + 6 \, a^{2} b^{2} c^{3} d^{7} - 4 \, a^{3} b c^{2} d^{8} + a^{4} c d^{9}\right )} x^{5} + 15 \,{\left (b^{4} c^{6} d^{4} - 4 \, a b^{3} c^{5} d^{5} + 6 \, a^{2} b^{2} c^{4} d^{6} - 4 \, a^{3} b c^{3} d^{7} + a^{4} c^{2} d^{8}\right )} x^{4} + 20 \,{\left (b^{4} c^{7} d^{3} - 4 \, a b^{3} c^{6} d^{4} + 6 \, a^{2} b^{2} c^{5} d^{5} - 4 \, a^{3} b c^{4} d^{6} + a^{4} c^{3} d^{7}\right )} x^{3} + 15 \,{\left (b^{4} c^{8} d^{2} - 4 \, a b^{3} c^{7} d^{3} + 6 \, a^{2} b^{2} c^{6} d^{4} - 4 \, a^{3} b c^{5} d^{5} + a^{4} c^{4} d^{6}\right )} x^{2} + 6 \,{\left (b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(37/6),x, algorithm="fricas")

[Out]

6/191425*(1296*b^5*d^3*x^5 + 14725*a^2*b^3*c^3 - 30225*a^3*b^2*c^2*d + 22971*a^4*b*c*d^2 - 6175*a^5*d^3 + 216*
(31*b^5*c*d^2 - a*b^4*d^3)*x^4 + 18*(775*b^5*c^2*d - 62*a*b^4*c*d^2 + 7*a^2*b^3*d^3)*x^3 + (14725*b^5*c^3 - 23
25*a*b^4*c^2*d + 651*a^2*b^3*c*d^2 - 91*a^3*b^2*d^3)*x^2 + 2*(14725*a*b^4*c^3 - 23250*a^2*b^3*c^2*d + 15717*a^
3*b^2*c*d^2 - 3952*a^4*b*d^3)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^4*c^10 - 4*a*b^3*c^9*d + 6*a^2*b^2*c^8*d^2
- 4*a^3*b*c^7*d^3 + a^4*c^6*d^4 + (b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^
10)*x^6 + 6*(b^4*c^5*d^5 - 4*a*b^3*c^4*d^6 + 6*a^2*b^2*c^3*d^7 - 4*a^3*b*c^2*d^8 + a^4*c*d^9)*x^5 + 15*(b^4*c^
6*d^4 - 4*a*b^3*c^5*d^5 + 6*a^2*b^2*c^4*d^6 - 4*a^3*b*c^3*d^7 + a^4*c^2*d^8)*x^4 + 20*(b^4*c^7*d^3 - 4*a*b^3*c
^6*d^4 + 6*a^2*b^2*c^5*d^5 - 4*a^3*b*c^4*d^6 + a^4*c^3*d^7)*x^3 + 15*(b^4*c^8*d^2 - 4*a*b^3*c^7*d^3 + 6*a^2*b^
2*c^6*d^4 - 4*a^3*b*c^5*d^5 + a^4*c^4*d^6)*x^2 + 6*(b^4*c^9*d - 4*a*b^3*c^8*d^2 + 6*a^2*b^2*c^7*d^3 - 4*a^3*b*
c^6*d^4 + a^4*c^5*d^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(7/6)/(d*x+c)**(37/6),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(37/6),x, algorithm="giac")

[Out]

Timed out