3.3004 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx\)

Optimal. Leaf size=409 \[ \frac{\log (a+b x) (-a d f-2 b c f+3 b d e)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{d} f^2}+\frac{\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac{3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f} \]

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/f + ((3*b*d*e - 2*b*c*f - a*d*f)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/
3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*b^(2/3)*d^(1/3)*f^2) - (Sqrt[3]*(b*e - a*f)^(1/3)*(d*e - c*f)
^(2/3)*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^2 + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[a + b*x])/(6*b^(2/3)*d^(1/3)*f^2) + ((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3
)*Log[e + f*x])/(2*f^2) - (3*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)*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^2) + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d
^(1/3)*(a + b*x)^(1/3))])/(2*b^(2/3)*d^(1/3)*f^2)

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Rubi [A]  time = 0.396485, antiderivative size = 409, 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 = {101, 157, 59, 91} \[ \frac{\log (a+b x) (-a d f-2 b c f+3 b d e)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{d} f^2}+\frac{\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac{3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/f + ((3*b*d*e - 2*b*c*f - a*d*f)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/
3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*b^(2/3)*d^(1/3)*f^2) - (Sqrt[3]*(b*e - a*f)^(1/3)*(d*e - c*f)
^(2/3)*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^2 + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[a + b*x])/(6*b^(2/3)*d^(1/3)*f^2) + ((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3
)*Log[e + f*x])/(2*f^2) - (3*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)*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^2) + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d
^(1/3)*(a + b*x)^(1/3))])/(2*b^(2/3)*d^(1/3)*f^2)

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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{\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac{\int \frac{\frac{1}{3} (b c e+2 a d e-3 a c f)+\frac{1}{3} (3 b d e-2 b c f-a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{f}\\ &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}+\frac{((b e-a f) (d e-c f)) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{f^2}-\frac{(3 b d e-2 b c f-a d f) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{3 f^2}\\ &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}+\frac{(3 b d e-2 b c f-a d f) \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 )}{\sqrt{3} b^{2/3} \sqrt [3]{d} f^2}-\frac{\sqrt{3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \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^2}+\frac{(3 b d e-2 b c f-a d f) \log (a+b x)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac{\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac{3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \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^2}+\frac{(3 b d e-2 b c f-a d f) \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 b^{2/3} \sqrt [3]{d} f^2}\\ \end{align*}

Mathematica [C]  time = 0.179292, size = 194, normalized size = 0.47 \[ \frac{3 \sqrt [3]{a+b x} \left (-\frac{d (b e-a f) \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{d (a+b x)}{a d-b c}\right )}{b f}+\frac{(d e-c f) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f}+\frac{(c+d x) \, _2F_1\left (-\frac{2}{3},\frac{1}{3};\frac{4}{3};\frac{d (a+b x)}{a d-b c}\right )}{\left (\frac{b (c+d x)}{b c-a d}\right )^{2/3}}\right )}{f \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(1/3)*(((c + d*x)*Hypergeometric2F1[-2/3, 1/3, 4/3, (d*(a + b*x))/(-(b*c) + a*d)])/((b*(c + d*x))
/(b*c - a*d))^(2/3) - (d*(b*e - a*f)*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (d*(a
+ b*x))/(-(b*c) + a*d)])/(b*f) + ((d*e - c*f)*Hypergeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a
*f)*(c + d*x))])/f))/(f*(c + d*x)^(1/3))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{fx+e}\sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*x+a)^(1/3)*(d*x+c)^(2/3)/(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{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 33.6434, size = 4668, normalized size = 11.41 \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)^(1/3)*(d*x+c)^(2/3)/(f*x+e),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{e + f x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x)**(1/3)*(c + d*x)**(2/3)/(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{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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