3.16.31 \(\int \frac {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac {432 b^2 (a+b x)^{13/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac {72 b (a+b x)^{13/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)} \]

________________________________________________________________________________________

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)^{13/6}}{6175 (c+d x)^{13/6} (b c-a d)^3}+\frac {72 b (a+b x)^{13/6}}{475 (c+d x)^{19/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{25 (c+d x)^{25/6} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/6)/(c + d*x)^(31/6),x]

[Out]

(6*(a + b*x)^(13/6))/(25*(b*c - a*d)*(c + d*x)^(25/6)) + (72*b*(a + b*x)^(13/6))/(475*(b*c - a*d)^2*(c + d*x)^
(19/6)) + (432*b^2*(a + b*x)^(13/6))/(6175*(b*c - a*d)^3*(c + d*x)^(13/6))

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 {(a+b x)^{7/6}}{(c+d x)^{31/6}} \, dx &=\frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {(12 b) \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx}{25 (b c-a d)}\\ &=\frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {\left (72 b^2\right ) \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{475 (b c-a d)^2}\\ &=\frac {6 (a+b x)^{13/6}}{25 (b c-a d) (c+d x)^{25/6}}+\frac {72 b (a+b x)^{13/6}}{475 (b c-a d)^2 (c+d x)^{19/6}}+\frac {432 b^2 (a+b x)^{13/6}}{6175 (b c-a d)^3 (c+d x)^{13/6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 77, normalized size = 0.76 \begin {gather*} \frac {6 (a+b x)^{13/6} \left (247 a^2 d^2-26 a b d (25 c+6 d x)+b^2 \left (475 c^2+300 c d x+72 d^2 x^2\right )\right )}{6175 (c+d x)^{25/6} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(7/6)/(c + d*x)^(31/6),x]

[Out]

(6*(a + b*x)^(13/6)*(247*a^2*d^2 - 26*a*b*d*(25*c + 6*d*x) + b^2*(475*c^2 + 300*c*d*x + 72*d^2*x^2)))/(6175*(b
*c - a*d)^3*(c + d*x)^(25/6))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.20, size = 73, normalized size = 0.72 \begin {gather*} \frac {6 (a+b x)^{13/6} \left (\frac {247 d^2 (a+b x)^2}{(c+d x)^2}-\frac {650 b d (a+b x)}{c+d x}+475 b^2\right )}{6175 (c+d x)^{13/6} (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(7/6)/(c + d*x)^(31/6),x]

[Out]

(6*(a + b*x)^(13/6)*(475*b^2 + (247*d^2*(a + b*x)^2)/(c + d*x)^2 - (650*b*d*(a + b*x))/(c + d*x)))/(6175*(b*c
- a*d)^3*(c + d*x)^(13/6))

________________________________________________________________________________________

fricas [B]  time = 1.37, size = 427, normalized size = 4.23 \begin {gather*} \frac {6 \, {\left (72 \, b^{4} d^{2} x^{4} + 475 \, a^{2} b^{2} c^{2} - 650 \, a^{3} b c d + 247 \, a^{4} d^{2} + 12 \, {\left (25 \, b^{4} c d - a b^{3} d^{2}\right )} x^{3} + {\left (475 \, b^{4} c^{2} - 50 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (475 \, a b^{3} c^{2} - 500 \, a^{2} b^{2} c d + 169 \, a^{3} b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{6175 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} x^{5} + 5 \, {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 10 \, {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{3} + 10 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2} + 5 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(31/6),x, algorithm="fricas")

[Out]

6/6175*(72*b^4*d^2*x^4 + 475*a^2*b^2*c^2 - 650*a^3*b*c*d + 247*a^4*d^2 + 12*(25*b^4*c*d - a*b^3*d^2)*x^3 + (47
5*b^4*c^2 - 50*a*b^3*c*d + 7*a^2*b^2*d^2)*x^2 + 2*(475*a*b^3*c^2 - 500*a^2*b^2*c*d + 169*a^3*b*d^2)*x)*(b*x +
a)^(1/6)*(d*x + c)^(5/6)/(b^3*c^8 - 3*a*b^2*c^7*d + 3*a^2*b*c^6*d^2 - a^3*c^5*d^3 + (b^3*c^3*d^5 - 3*a*b^2*c^2
*d^6 + 3*a^2*b*c*d^7 - a^3*d^8)*x^5 + 5*(b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3*c*d^7)*x^4 + 10
*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - a^3*c^2*d^6)*x^3 + 10*(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a
^2*b*c^4*d^4 - a^3*c^3*d^5)*x^2 + 5*(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c^4*d^4)*x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(31/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(31/6), x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 105, normalized size = 1.04 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (72 b^{2} x^{2} d^{2}-156 a b \,d^{2} x +300 b^{2} c d x +247 a^{2} d^{2}-650 a b c d +475 b^{2} c^{2}\right )}{6175 \left (d x +c \right )^{\frac {25}{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((b*x+a)^(7/6)/(d*x+c)^(31/6),x)

[Out]

-6/6175*(b*x+a)^(13/6)*(72*b^2*d^2*x^2-156*a*b*d^2*x+300*b^2*c*d*x+247*a^2*d^2-650*a*b*c*d+475*b^2*c^2)/(d*x+c
)^(25/6)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {31}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(31/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(31/6), x)

________________________________________________________________________________________

mupad [B]  time = 1.14, size = 278, normalized size = 2.75 \begin {gather*} -\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {{\left (a+b\,x\right )}^{1/6}\,\left (1482\,a^4\,d^2-3900\,a^3\,b\,c\,d+2850\,a^2\,b^2\,c^2\right )}{6175\,d^5\,{\left (a\,d-b\,c\right )}^3}+\frac {432\,b^4\,x^4\,{\left (a+b\,x\right )}^{1/6}}{6175\,d^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (42\,a^2\,b^2\,d^2-300\,a\,b^3\,c\,d+2850\,b^4\,c^2\right )}{6175\,d^5\,{\left (a\,d-b\,c\right )}^3}-\frac {72\,b^3\,x^3\,\left (a\,d-25\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{6175\,d^4\,{\left (a\,d-b\,c\right )}^3}+\frac {12\,a\,b\,x\,{\left (a+b\,x\right )}^{1/6}\,\left (169\,a^2\,d^2-500\,a\,b\,c\,d+475\,b^2\,c^2\right )}{6175\,d^5\,{\left (a\,d-b\,c\right )}^3}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(7/6)/(c + d*x)^(31/6),x)

[Out]

-((c + d*x)^(5/6)*(((a + b*x)^(1/6)*(1482*a^4*d^2 + 2850*a^2*b^2*c^2 - 3900*a^3*b*c*d))/(6175*d^5*(a*d - b*c)^
3) + (432*b^4*x^4*(a + b*x)^(1/6))/(6175*d^3*(a*d - b*c)^3) + (x^2*(a + b*x)^(1/6)*(2850*b^4*c^2 + 42*a^2*b^2*
d^2 - 300*a*b^3*c*d))/(6175*d^5*(a*d - b*c)^3) - (72*b^3*x^3*(a*d - 25*b*c)*(a + b*x)^(1/6))/(6175*d^4*(a*d -
b*c)^3) + (12*a*b*x*(a + b*x)^(1/6)*(169*a^2*d^2 + 475*b^2*c^2 - 500*a*b*c*d))/(6175*d^5*(a*d - b*c)^3)))/(x^5
+ c^5/d^5 + (5*c*x^4)/d + (5*c^4*x)/d^4 + (10*c^2*x^3)/d^2 + (10*c^3*x^2)/d^3)

________________________________________________________________________________________

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)**(7/6)/(d*x+c)**(31/6),x)

[Out]

Timed out

________________________________________________________________________________________