3.31.41 \(\int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx\) [3041]

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

[Out]

-3*(b*x+a)^(4/3)/(-c*f+d*e)/(d*x+c)^(1/3)/(f*x+e)+4*(-a*f+b*e)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e)^2/(f*x+e
)-2/3*(-a*d+b*c)*(-a*f+b*e)^(1/3)*ln(f*x+e)/(-c*f+d*e)^(7/3)+2*(-a*d+b*c)*(-a*f+b*e)^(1/3)*ln(-(b*x+a)^(1/3)+(
-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3))/(-c*f+d*e)^(7/3)+4/3*(-a*d+b*c)*(-a*f+b*e)^(1/3)*arctan(1/3*3^
(1/2)+2/3*(-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3)/(b*x+a)^(1/3)*3^(1/2))/(-c*f+d*e)^(7/3)*3^(1/2)

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Rubi [A]
time = 0.11, 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 = {96, 93} \begin {gather*} \frac {4 (b c-a d) \sqrt [3]{b e-a f} \text {ArcTan}\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}}-\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}} \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 93

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[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] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.05, size = 123, normalized size = 0.41 \begin {gather*} \frac {\sqrt [3]{a+b x} \left (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) \, _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 )\right )}{(d e-c f)^2 \sqrt [3]{c+d x} (e+f x)} \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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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{2}}\, dx\]

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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (275) = 550\).
time = 1.79, size = 662, normalized size = 2.20 \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 )} f x + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (c f - d e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} + \sqrt {3} {\left (a d f x + a c f - {\left (b d x + b c\right )} e\right )}}{3 \, {\left (a d f x + a c f - {\left (b d x + b c\right )} e\right )}}\right ) + 2 \, {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} f x + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {a f - b e}{c f - d e}\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 )} f x + {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )} e\right )} \left (-\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {a f - b e}{c f - d e}\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 (3 \, b c - 4 \, a d\right )} f x - {\left (b d x + 4 \, b c - 3 \, a d\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{3 \, {\left (c^{2} d f^{3} x^{2} + c^{3} f^{3} x + {\left (d^{3} x + c d^{2}\right )} e^{3} + {\left (d^{3} f x^{2} - c d^{2} f x - 2 \, c^{2} d f\right )} e^{2} - {\left (2 \, c d^{2} f^{2} x^{2} + c^{2} d f^{2} x - c^{3} f^{2}\right )} e\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)*f*x + (b*c^2 - a*c*d + (b*c*d - a*d^2)*x)*e)*(-(a*f -
b*e)/(c*f - d*e))^(1/3)*arctan(1/3*(2*sqrt(3)*(c*f - d*e)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*(-(a*f - b*e)/(c*f -
d*e))^(2/3) + sqrt(3)*(a*d*f*x + a*c*f - (b*d*x + b*c)*e))/(a*d*f*x + a*c*f - (b*d*x + b*c)*e)) + 2*((b*c*d -
a*d^2)*f*x^2 + (b*c^2 - a*c*d)*f*x + (b*c^2 - a*c*d + (b*c*d - a*d^2)*x)*e)*(-(a*f - b*e)/(c*f - d*e))^(1/3)*
log(((d*x + c)*(-(a*f - b*e)/(c*f - d*e))^(2/3) - (b*x + a)^(1/3)*(d*x + c)^(2/3)*(-(a*f - b*e)/(c*f - d*e))^(
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)*f*x + (b*c^2 -
a*c*d + (b*c*d - a*d^2)*x)*e)*(-(a*f - b*e)/(c*f - d*e))^(1/3)*log(((d*x + c)*(-(a*f - b*e)/(c*f - d*e))^(1/3
) + (b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 3*(a*c*f - (3*b*c - 4*a*d)*f*x - (b*d*x + 4*b*c - 3*a*d)*e)*
(b*x + a)^(1/3)*(d*x + c)^(2/3))/(c^2*d*f^3*x^2 + c^3*f^3*x + (d^3*x + c*d^2)*e^3 + (d^3*f*x^2 - c*d^2*f*x - 2
*c^2*d*f)*e^2 - (2*c*d^2*f^2*x^2 + c^2*d*f^2*x - c^3*f^2)*e)

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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