3.3040 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx\)

Optimal. Leaf size=380 \[ -\frac{3 b^{4/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3} f}-\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{d^{4/3} f}-\frac{b^{4/3} \log (a+b x)}{2 d^{4/3} f}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (d e-c f)}-\frac{(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac{3 (b e-a f)^{4/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f (d e-c f)^{4/3}}+\frac{\sqrt{3} (b e-a f)^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f (d e-c f)^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.350777, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {98, 157, 59, 91} \[ -\frac{3 b^{4/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3} f}-\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{d^{4/3} f}-\frac{b^{4/3} \log (a+b x)}{2 d^{4/3} f}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (d e-c f)}-\frac{(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac{3 (b e-a f)^{4/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f (d e-c f)^{4/3}}+\frac{\sqrt{3} (b e-a f)^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f (d e-c f)^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 98

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

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

Mathematica [C]  time = 0.219061, size = 148, normalized size = 0.39 \[ \frac{3 \sqrt [3]{a+b x} \left (\frac{(a+b x) \left (\frac{b (c+d x)}{b c-a d}\right )^{4/3} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{d (a+b x)}{a d-b c}\right )}{c+d x}-\frac{4 (b e-a f) \left (\, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )-1\right )}{d e-c f}\right )}{4 f \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(1/3)*((-4*(b*e - a*f)*(-1 + Hypergeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*
(c + d*x))]))/(d*e - c*f) + ((a + b*x)*((b*(c + d*x))/(b*c - a*d))^(4/3)*Hypergeometric2F1[4/3, 4/3, 7/3, (d*(
a + b*x))/(-(b*c) + a*d)])/(c + d*x)))/(4*f*(c + d*x)^(1/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{fx+e} \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),x)

[Out]

int((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e),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 )}}\,{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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 3.54272, size = 1727, normalized size = 4.54 \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),x, algorithm="fricas")

[Out]

1/2*(6*(b*c - a*d)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*f - 2*sqrt(3)*(b*c*d*e - a*c*d*f + (b*d^2*e - a*d^2*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*sqr
t(3)*(b*c*d*e - b*c^2*f + (b*d^2*e - b*c*d*f)*x)*(-b/d)^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x + a)^(1/3)*(d*x + c)^
(2/3)*d*(-b/d)^(2/3) + sqrt(3)*(b*d*x + b*c))/(b*d*x + b*c)) - (b*c*d*e - a*c*d*f + (b*d^2*e - a*d^2*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)) - (b*c*d*e - b*c^2*f + (b*d^2*e
- b*c*d*f)*x)*(-b/d)^(1/3)*log(((d*x + c)*(-b/d)^(2/3) - (b*x + a)^(1/3)*(d*x + c)^(2/3)*(-b/d)^(1/3) + (b*x +
a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) + 2*(b*c*d*e - a*c*d*f + (b*d^2*e - a*d^2*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)) + 2*(b*c
*d*e - b*c^2*f + (b*d^2*e - b*c*d*f)*x)*(-b/d)^(1/3)*log(((d*x + c)*(-b/d)^(1/3) + (b*x + a)^(1/3)*(d*x + c)^(
2/3))/(d*x + c)))/(c*d^2*e*f - c^2*d*f^2 + (d^3*e*f - c*d^2*f^2)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{4}{3}}}{\left (c + d x\right )^{\frac{4}{3}} \left (e + f x\right )}\, dx \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),x)

[Out]

Integral((a + b*x)**(4/3)/((c + d*x)**(4/3)*(e + f*x)), x)

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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 )}}\,{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),x, algorithm="giac")

[Out]

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