3.26.26 \(\int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx\)

Optimal. Leaf size=301 \[ -\frac {3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}+\frac {4 \sqrt [3]{a+b x} (c+d x)^{2/3} (b e-a f)}{(e+f x) (d e-c f)^2}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{(d e-c f)^{7/3}}+\frac {4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} (d e-c f)^{7/3}} \]

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Rubi [A]  time = 0.15, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {94, 91} \begin {gather*} -\frac {3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}+\frac {4 \sqrt [3]{a+b x} (c+d x)^{2/3} (b e-a f)}{(e+f x) (d e-c f)^2}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{(d e-c f)^{7/3}}+\frac {4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} (d e-c f)^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(4/3)/((c + d*x)^(4/3)*(e + f*x)^2),x]

[Out]

(-3*(a + b*x)^(4/3))/((d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)) + (4*(b*e - a*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))
/((d*e - c*f)^2*(e + f*x)) + (4*(b*c - a*d)*(b*e - a*f)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x
)^(1/3))/(Sqrt[3]*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*(d*e - c*f)^(7/3)) - (2*(b*c - a*d)*(b*e - a*f
)^(1/3)*Log[e + f*x])/(3*(d*e - c*f)^(7/3)) + (2*(b*c - a*d)*(b*e - a*f)^(1/3)*Log[-(a + b*x)^(1/3) + ((b*e -
a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(d*e - c*f)^(7/3)

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx &=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {(4 (b e-a f)) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx}{d e-c f}\\ &=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {4 (b e-a f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f)^2 (e+f x)}-\frac {(4 (b c-a d) (b e-a f)) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{3 (d e-c f)^2}\\ &=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {4 (b e-a f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f)^2 (e+f x)}+\frac {4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} (d e-c f)^{7/3}}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{(d e-c f)^{7/3}}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 123, normalized size = 0.41 \begin {gather*} \frac {\sqrt [3]{a+b x} \left (-4 (e+f x) (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )-a (c f+3 d e+4 d f x)+b (4 c e+3 c f x+d e x)\right )}{\sqrt [3]{c+d x} (e+f x) (d e-c f)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(4/3)/((c + d*x)^(4/3)*(e + f*x)^2),x]

[Out]

((a + b*x)^(1/3)*(b*(4*c*e + d*e*x + 3*c*f*x) - a*(3*d*e + c*f + 4*d*f*x) - 4*(b*c - a*d)*(e + f*x)*Hypergeome
tric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/((d*e - c*f)^2*(c + d*x)^(1/3)*(e + f*
x))

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IntegrateAlgebraic [B]  time = 30.24, size = 759, normalized size = 2.52 \begin {gather*} -\frac {\sqrt [3]{d} \sqrt [3]{a+b x} \left (\sqrt [3]{d} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}+\sqrt [3]{a d+b d x} \sqrt [3]{c f-d e}\right ) \left (d^{2/3} (c+d x)^{2/3} (b e-a f)^{2/3}-\sqrt [3]{d} \sqrt [3]{c+d x} \sqrt [3]{a d+b d x} \sqrt [3]{b e-a f} \sqrt [3]{c f-d e}+(a d+b d x)^{2/3} (c f-d e)^{2/3}\right ) \left (-\frac {\sqrt [3]{a d+b (c+d x)-b c} \left (4 a d f (c+d x)-3 a c d f+3 a d^2 e+3 b c^2 f-b d e (c+d x)-3 b c d e-3 b c f (c+d x)\right )}{\sqrt [3]{d} \sqrt [3]{c+d x} (d e-c f)^2 (f (c+d x)-c f+d e)}+\frac {2 \left (b c \sqrt [3]{b e-a f}-a d \sqrt [3]{b e-a f}\right ) \log \left (d^{2/3} (c+d x)^{2/3} (b e-a f)^{2/3}-\sqrt [3]{d} \sqrt [3]{c+d x} \sqrt [3]{b e-a f} \sqrt [3]{c f-d e} \sqrt [3]{a d+b (c+d x)-b c}+(c f-d e)^{2/3} (a d+b (c+d x)-b c)^{2/3}\right )}{3 (c f-d e)^{7/3}}-\frac {4 \left (b c \sqrt [3]{b e-a f}-a d \sqrt [3]{b e-a f}\right ) \log \left (\sqrt [3]{d} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}+\sqrt [3]{c f-d e} \sqrt [3]{a d+b (c+d x)-b c}\right )}{3 (c f-d e)^{7/3}}-\frac {4 \left (\sqrt {3} b c \sqrt [3]{b e-a f}-\sqrt {3} a d \sqrt [3]{b e-a f}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}-2 \sqrt [3]{c f-d e} \sqrt [3]{a d+b (c+d x)-b c}}\right )}{3 (c f-d e)^{7/3}}\right )}{(b c-a d) \sqrt [3]{a d+b d x} (-d e-d f x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(4/3)/((c + d*x)^(4/3)*(e + f*x)^2),x]

[Out]

-((d^(1/3)*(a + b*x)^(1/3)*(d^(1/3)*(b*e - a*f)^(1/3)*(c + d*x)^(1/3) + (-(d*e) + c*f)^(1/3)*(a*d + b*d*x)^(1/
3))*(d^(2/3)*(b*e - a*f)^(2/3)*(c + d*x)^(2/3) - d^(1/3)*(b*e - a*f)^(1/3)*(-(d*e) + c*f)^(1/3)*(c + d*x)^(1/3
)*(a*d + b*d*x)^(1/3) + (-(d*e) + c*f)^(2/3)*(a*d + b*d*x)^(2/3))*(-(((-(b*c) + a*d + b*(c + d*x))^(1/3)*(-3*b
*c*d*e + 3*a*d^2*e + 3*b*c^2*f - 3*a*c*d*f - b*d*e*(c + d*x) - 3*b*c*f*(c + d*x) + 4*a*d*f*(c + d*x)))/(d^(1/3
)*(d*e - c*f)^2*(c + d*x)^(1/3)*(d*e - c*f + f*(c + d*x)))) - (4*(Sqrt[3]*b*c*(b*e - a*f)^(1/3) - Sqrt[3]*a*d*
(b*e - a*f)^(1/3))*ArcTan[(Sqrt[3]*d^(1/3)*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(b*e - a*f)^(1/3)*(c +
d*x)^(1/3) - 2*(-(d*e) + c*f)^(1/3)*(-(b*c) + a*d + b*(c + d*x))^(1/3))])/(3*(-(d*e) + c*f)^(7/3)) - (4*(b*c*(
b*e - a*f)^(1/3) - a*d*(b*e - a*f)^(1/3))*Log[d^(1/3)*(b*e - a*f)^(1/3)*(c + d*x)^(1/3) + (-(d*e) + c*f)^(1/3)
*(-(b*c) + a*d + b*(c + d*x))^(1/3)])/(3*(-(d*e) + c*f)^(7/3)) + (2*(b*c*(b*e - a*f)^(1/3) - a*d*(b*e - a*f)^(
1/3))*Log[d^(2/3)*(b*e - a*f)^(2/3)*(c + d*x)^(2/3) - d^(1/3)*(b*e - a*f)^(1/3)*(-(d*e) + c*f)^(1/3)*(c + d*x)
^(1/3)*(-(b*c) + a*d + b*(c + d*x))^(1/3) + (-(d*e) + c*f)^(2/3)*(-(b*c) + a*d + b*(c + d*x))^(2/3)])/(3*(-(d*
e) + c*f)^(7/3))))/((b*c - a*d)*(a*d + b*d*x)^(1/3)*(-(d*e) - d*f*x)))

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fricas [B]  time = 1.35, size = 651, normalized size = 2.16 \begin {gather*} \frac {4 \, \sqrt {3} {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} e + {\left ({\left (b c d - a d^{2}\right )} e + {\left (b c^{2} - a c d\right )} f\right )} x\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (d e - c f\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b c e - a c f + {\left (b d e - a d f\right )} x\right )}}{3 \, {\left (b c e - a c f + {\left (b d e - a d f\right )} x\right )}}\right ) + 2 \, {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} e + {\left ({\left (b c d - a d^{2}\right )} e + {\left (b c^{2} - a c d\right )} f\right )} x\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 4 \, {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} e + {\left ({\left (b c d - a d^{2}\right )} e + {\left (b c^{2} - a c d\right )} f\right )} x\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) - 3 \, {\left (a c f - {\left (4 \, b c - 3 \, a d\right )} e - {\left (b d e + {\left (3 \, b c - 4 \, a d\right )} f\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{3 \, {\left (c d^{2} e^{3} - 2 \, c^{2} d e^{2} f + c^{3} e f^{2} + {\left (d^{3} e^{2} f - 2 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} x^{2} + {\left (d^{3} e^{3} - c d^{2} e^{2} f - c^{2} d e f^{2} + c^{3} f^{3}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^2,x, algorithm="fricas")

[Out]

1/3*(4*sqrt(3)*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a*d^2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e
- a*f)/(d*e - c*f))^(1/3)*arctan(1/3*(2*sqrt(3)*(d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*(-(b*e - a*f)/(d*e
- c*f))^(2/3) + sqrt(3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(b*c*e - a*c*f + (b*d*e - a*d*f)*x)) + 2*((b*c*d
- a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a*d^2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1
/3)*log(((d*x + c)*(-(b*e - a*f)/(d*e - c*f))^(2/3) - (b*x + a)^(1/3)*(d*x + c)^(2/3)*(-(b*e - a*f)/(d*e - c*f
))^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) - 4*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*
d - a*d^2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)*log(((d*x + c)*(-(b*e - a*f)/(d*e - c*f)
)^(1/3) + (b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 3*(a*c*f - (4*b*c - 3*a*d)*e - (b*d*e + (3*b*c - 4*a*d
)*f)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(c*d^2*e^3 - 2*c^2*d*e^2*f + c^3*e*f^2 + (d^3*e^2*f - 2*c*d^2*e*f^2 +
c^2*d*f^3)*x^2 + (d^3*e^3 - c*d^2*e^2*f - c^2*d*e*f^2 + c^3*f^3)*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 {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^(4/3)/((d*x + c)^(4/3)*(f*x + e)^2), x)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^2,x)

[Out]

int((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^2,x)

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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 {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(4/3)/((d*x + c)^(4/3)*(f*x + e)^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{4/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(4/3)/((e + f*x)^2*(c + d*x)^(4/3)),x)

[Out]

int((a + b*x)^(4/3)/((e + f*x)^2*(c + d*x)^(4/3)), x)

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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)**(4/3)/(d*x+c)**(4/3)/(f*x+e)**2,x)

[Out]

Timed out

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