3.3041 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx\)

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

[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)

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Rubi [A]  time = 0.149254, antiderivative size = 301, normalized size of antiderivative = 1., 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} \[ -\frac{3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}+\frac{4 \sqrt [3]{a+b x} (c+d x)^{2/3} (b e-a f)}{(e+f x) (d e-c f)^2}-\frac{2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac{2 (b c-a d) \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{(d e-c f)^{7/3}}+\frac{4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (d e-c f)^{7/3}} \]

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 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])

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]

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*}

Mathematica [C]  time = 0.0548897, size = 123, normalized size = 0.41 \[ \frac{\sqrt [3]{a+b x} \left (-4 (e+f x) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )-a (c f+3 d e+4 d f x)+b (4 c e+3 c f x+d e x)\right )}{\sqrt [3]{c+d x} (e+f x) (d e-c f)^2} \]

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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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{2}} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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{align*}

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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Fricas [B]  time = 1.91665, size = 1431, normalized size = 4.75 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)