\(\int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx\) [1605]

Optimal result

Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=-\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}+\frac {9 d \sqrt [3]{c+d x}}{4 (b c-a d)^2 \sqrt [3]{a+b x}} \]

[Out]

-3/4*(d*x+c)^(1/3)/(-a*d+b*c)/(b*x+a)^(4/3)+9/4*d*(d*x+c)^(1/3)/(-a*d+b*c)^2/(b*x+a)^(1/3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=\frac {9 d \sqrt [3]{c+d x}}{4 \sqrt [3]{a+b x} (b c-a d)^2}-\frac {3 \sqrt [3]{c+d x}}{4 (a+b x)^{4/3} (b c-a d)} \]

[In]

Int[1/((a + b*x)^(7/3)*(c + d*x)^(2/3)),x]

[Out]

(-3*(c + d*x)^(1/3))/(4*(b*c - a*d)*(a + b*x)^(4/3)) + (9*d*(c + d*x)^(1/3))/(4*(b*c - a*d)^2*(a + b*x)^(1/3))

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 47

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*} \text {integral}& = -\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}-\frac {(3 d) \int \frac {1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx}{4 (b c-a d)} \\ & = -\frac {3 \sqrt [3]{c+d x}}{4 (b c-a d) (a+b x)^{4/3}}+\frac {9 d \sqrt [3]{c+d x}}{4 (b c-a d)^2 \sqrt [3]{a+b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=-\frac {3 \sqrt [3]{c+d x} (b c-4 a d-3 b d x)}{4 (b c-a d)^2 (a+b x)^{4/3}} \]

[In]

Integrate[1/((a + b*x)^(7/3)*(c + d*x)^(2/3)),x]

[Out]

(-3*(c + d*x)^(1/3)*(b*c - 4*a*d - 3*b*d*x))/(4*(b*c - a*d)^2*(a + b*x)^(4/3))

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82

method	result	size
gosper	\(\frac {3 \left (d x +c \right )^{\frac {1}{3}} \left (3 b d x +4 a d -b c \right )}{4 \left (b x +a \right )^{\frac {4}{3}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(54\)

[In]

int(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3/4*(d*x+c)^(1/3)*(3*b*d*x+4*a*d-b*c)/(b*x+a)^(4/3)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).

Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=\frac {3 \, {\left (3 \, b d x - b c + 4 \, a d\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]

[In]

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x, algorithm="fricas")

[Out]

3/4*(3*b*d*x - b*c + 4*a*d)*(b*x + a)^(2/3)*(d*x + c)^(1/3)/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 -
2*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x)

Sympy [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {7}{3}} \left (c + d x\right )^{\frac {2}{3}}}\, dx \]

[In]

integrate(1/(b*x+a)**(7/3)/(d*x+c)**(2/3),x)

[Out]

Integral(1/((a + b*x)**(7/3)*(c + d*x)**(2/3)), x)

Maxima [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/3)*(d*x + c)^(2/3)), x)

Giac [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/3)*(d*x + c)^(2/3)), x)

Mupad [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{2/3}} \, dx=\frac {\left (\frac {9\,d\,x}{4\,{\left (a\,d-b\,c\right )}^2}+\frac {12\,a\,d-3\,b\,c}{4\,b\,{\left (a\,d-b\,c\right )}^2}\right )\,{\left (c+d\,x\right )}^{1/3}}{x\,{\left (a+b\,x\right )}^{1/3}+\frac {a\,{\left (a+b\,x\right )}^{1/3}}{b}} \]

[In]

int(1/((a + b*x)^(7/3)*(c + d*x)^(2/3)),x)

[Out]

(((9*d*x)/(4*(a*d - b*c)^2) + (12*a*d - 3*b*c)/(4*b*(a*d - b*c)^2))*(c + d*x)^(1/3))/(x*(a + b*x)^(1/3) + (a*(
a + b*x)^(1/3))/b)