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

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

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Rubi [A]  time = 0.03, antiderivative size = 136, 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^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)} \end {gather*}

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 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)^{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*}

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Mathematica [A]  time = 0.09, size = 118, normalized size = 0.87 \begin {gather*} \frac {6 (a+b x)^{13/6} \left (-6175 a^3 d^3+741 a^2 b d^2 (31 c+6 d x)-39 a b^2 d \left (775 c^2+372 c d x+72 d^2 x^2\right )+b^3 \left (14725 c^3+13950 c^2 d x+6696 c d^2 x^2+1296 d^3 x^3\right )\right )}{191425 (c+d x)^{31/6} (b c-a d)^4} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.23, size = 95, normalized size = 0.70 \begin {gather*} \frac {6 (a+b x)^{13/6} \left (-\frac {30225 b^2 d (a+b x)}{c+d x}-\frac {6175 d^3 (a+b x)^3}{(c+d x)^3}+\frac {22971 b d^2 (a+b x)^2}{(c+d x)^2}+14725 b^3\right )}{191425 (c+d x)^{13/6} (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(13/6)*(14725*b^3 - (6175*d^3*(a + b*x)^3)/(c + d*x)^3 + (22971*b*d^2*(a + b*x)^2)/(c + d*x)^2 -
(30225*b^2*d*(a + b*x))/(c + d*x)))/(191425*(b*c - a*d)^4*(c + d*x)^(13/6))

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fricas [B]  time = 1.39, size = 649, normalized size = 4.77 \begin {gather*} \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 {gather*}

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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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 {37}{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)^(37/6),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 171, normalized size = 1.26 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (-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}\right )}{191425 \left (d x +c \right )^{\frac {31}{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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {37}{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)^(37/6),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.43, size = 385, normalized size = 2.83 \begin {gather*} \frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {7776\,b^5\,x^5\,{\left (a+b\,x\right )}^{1/6}}{191425\,d^3\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{1/6}\,\left (37050\,a^5\,d^3-137826\,a^4\,b\,c\,d^2+181350\,a^3\,b^2\,c^2\,d-88350\,a^2\,b^3\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x^2\,{\left (a+b\,x\right )}^{1/6}\,\left (-546\,a^3\,b^2\,d^3+3906\,a^2\,b^3\,c\,d^2-13950\,a\,b^4\,c^2\,d+88350\,b^5\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{1/6}\,\left (-47424\,a^4\,b\,d^3+188604\,a^3\,b^2\,c\,d^2-279000\,a^2\,b^3\,c^2\,d+176700\,a\,b^4\,c^3\right )}{191425\,d^6\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^3\,x^3\,{\left (a+b\,x\right )}^{1/6}\,\left (7\,a^2\,d^2-62\,a\,b\,c\,d+775\,b^2\,c^2\right )}{191425\,d^5\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^4\,x^4\,\left (a\,d-31\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{191425\,d^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c + d*x)^(5/6)*((7776*b^5*x^5*(a + b*x)^(1/6))/(191425*d^3*(a*d - b*c)^4) - ((a + b*x)^(1/6)*(37050*a^5*d^3
- 88350*a^2*b^3*c^3 + 181350*a^3*b^2*c^2*d - 137826*a^4*b*c*d^2))/(191425*d^6*(a*d - b*c)^4) + (x^2*(a + b*x)^
(1/6)*(88350*b^5*c^3 - 546*a^3*b^2*d^3 + 3906*a^2*b^3*c*d^2 - 13950*a*b^4*c^2*d))/(191425*d^6*(a*d - b*c)^4) +
(x*(a + b*x)^(1/6)*(176700*a*b^4*c^3 - 47424*a^4*b*d^3 - 279000*a^2*b^3*c^2*d + 188604*a^3*b^2*c*d^2))/(19142
5*d^6*(a*d - b*c)^4) + (108*b^3*x^3*(a + b*x)^(1/6)*(7*a^2*d^2 + 775*b^2*c^2 - 62*a*b*c*d))/(191425*d^5*(a*d -
b*c)^4) - (1296*b^4*x^4*(a*d - 31*b*c)*(a + b*x)^(1/6))/(191425*d^4*(a*d - b*c)^4)))/(x^6 + c^6/d^6 + (6*c*x^
5)/d + (6*c^5*x)/d^5 + (15*c^2*x^4)/d^2 + (20*c^3*x^3)/d^3 + (15*c^4*x^2)/d^4)

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

[Out]

Timed out

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