3.31.38 \(\int \frac {(a+b x)^{4/3} (e+f x)}{(c+d x)^{4/3}} \, dx\) [3038]

Optimal. Leaf size=328 \[ \frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}+\frac {2 (6 b d e-7 b c f+a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}+\frac {2 (b c-a d) (6 b d e-7 b c f+a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log (a+b x)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 b^{2/3} d^{10/3}} \]

[Out]

3*(-c*f+d*e)*(b*x+a)^(7/3)/d/(-a*d+b*c)/(d*x+c)^(1/3)+2/3*(a*d*f-7*b*c*f+6*b*d*e)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/
d^3-1/2*(a*d*f-7*b*c*f+6*b*d*e)*(b*x+a)^(4/3)*(d*x+c)^(2/3)/d^2/(-a*d+b*c)+1/9*(-a*d+b*c)*(a*d*f-7*b*c*f+6*b*d
*e)*ln(b*x+a)/b^(2/3)/d^(10/3)+1/3*(-a*d+b*c)*(a*d*f-7*b*c*f+6*b*d*e)*ln(-1+b^(1/3)*(d*x+c)^(1/3)/d^(1/3)/(b*x
+a)^(1/3))/b^(2/3)/d^(10/3)+2/9*(-a*d+b*c)*(a*d*f-7*b*c*f+6*b*d*e)*arctan(1/3*3^(1/2)+2/3*b^(1/3)*(d*x+c)^(1/3
)/d^(1/3)/(b*x+a)^(1/3)*3^(1/2))/b^(2/3)/d^(10/3)*3^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 52, 61} \begin {gather*} \frac {2 (b c-a d) \text {ArcTan}\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 ) (a d f-7 b c f+6 b d e)}{3 \sqrt {3} b^{2/3} d^{10/3}}+\frac {(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac {2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}-\frac {(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac {3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(d*e - c*f)*(a + b*x)^(7/3))/(d*(b*c - a*d)*(c + d*x)^(1/3)) + (2*(6*b*d*e - 7*b*c*f + a*d*f)*(a + b*x)^(1/
3)*(c + d*x)^(2/3))/(3*d^3) - ((6*b*d*e - 7*b*c*f + a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*d^2*(b*c - a*d)
) + (2*(b*c - a*d)*(6*b*d*e - 7*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))])/(3*Sqrt[3]*b^(2/3)*d^(10/3)) + ((b*c - a*d)*(6*b*d*e - 7*b*c*f + a*d*f)*Log[a + b*x])/(9*b
^(2/3)*d^(10/3)) + ((b*c - a*d)*(6*b*d*e - 7*b*c*f + a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b
*x)^(1/3))])/(3*b^(2/3)*d^(10/3))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 61

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

Rule 79

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

Rubi steps

\begin {align*} \int \frac {(a+b x)^{4/3} (e+f x)}{(c+d x)^{4/3}} \, dx &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}-\frac {(6 b d e-7 b c f+a d f) \int \frac {(a+b x)^{4/3}}{\sqrt [3]{c+d x}} \, dx}{d (b c-a d)}\\ &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}+\frac {(2 (6 b d e-7 b c f+a d f)) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{3 d^2}\\ &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}+\frac {2 (6 b d e-7 b c f+a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}-\frac {(2 (b c-a d) (6 b d e-7 b c f+a d f)) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 d^3}\\ &=\frac {3 (d e-c f) (a+b x)^{7/3}}{d (b c-a d) \sqrt [3]{c+d x}}+\frac {2 (6 b d e-7 b c f+a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^3}-\frac {(6 b d e-7 b c f+a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 d^2 (b c-a d)}+\frac {2 (b c-a d) (6 b d e-7 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 )}{3 \sqrt {3} b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 b c f+a d f) \log (a+b x)}{9 b^{2/3} d^{10/3}}+\frac {(b c-a d) (6 b d e-7 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 )}{3 b^{2/3} d^{10/3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.12, size = 114, normalized size = 0.35 \begin {gather*} \frac {3 (a+b x)^{7/3} \left (b (-7 d e+7 c f)+(6 b d e-7 b c f+a d f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{3},\frac {7}{3};\frac {10}{3};\frac {d (a+b x)}{-b c+a d}\right )\right )}{7 b d (-b c+a d) \sqrt [3]{c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(7/3)*(b*(-7*d*e + 7*c*f) + (6*b*d*e - 7*b*c*f + a*d*f)*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hyperge
ometric2F1[1/3, 7/3, 10/3, (d*(a + b*x))/(-(b*c) + a*d)]))/(7*b*d*(-(b*c) + a*d)*(c + d*x)^(1/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}} \left (f x +e \right )}{\left (d x +c \right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (285) = 570\).
time = 1.72, size = 1358, normalized size = 4.14 \begin {gather*} \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, b^{3} c^{2} d^{2} - 8 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} f x + {\left (7 \, b^{3} c^{3} d - 8 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} f - 6 \, {\left (b^{3} c^{2} d^{2} - a b^{2} c d^{3} + {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x\right )} e\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (3 \, b^{2} d x + b^{2} c + 2 \, a b d + 3 \, \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}\right ) + 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 4 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{3} f x^{2} - 7 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f x - {\left (28 \, b^{3} c^{2} d - 25 \, a b^{2} c d^{2}\right )} f + 6 \, {\left (b^{3} d^{3} x + 4 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}, \frac {12 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, b^{3} c^{2} d^{2} - 8 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} f x + {\left (7 \, b^{3} c^{3} d - 8 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} f - 6 \, {\left (b^{3} c^{2} d^{2} - a b^{2} c d^{3} + {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x\right )} e\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}}{b^{2} d x + b^{2} c}\right ) + 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 4 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f x + {\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f - 6 \, {\left (b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{3} f x^{2} - 7 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f x - {\left (28 \, b^{3} c^{2} d - 25 \, a b^{2} c d^{2}\right )} f + 6 \, {\left (b^{3} d^{3} x + 4 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/18*(6*sqrt(1/3)*((7*b^3*c^2*d^2 - 8*a*b^2*c*d^3 + a^2*b*d^4)*f*x + (7*b^3*c^3*d - 8*a*b^2*c^2*d^2 + a^2*b*c
*d^3)*f - 6*(b^3*c^2*d^2 - a*b^2*c*d^3 + (b^3*c*d^3 - a*b^2*d^4)*x)*e)*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)) + 2*(-b^2*d)^(2/3)*((7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*f*x + (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f
- 6*(b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x)*e)*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)) - 4*(-b^2*d)^(2/3)*((7*b^2*c^
2*d - 8*a*b*c*d^2 + a^2*d^3)*f*x + (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f - 6*(b^2*c^2*d - a*b*c*d^2 + (b^2*c
*d^2 - a*b*d^3)*x)*e)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (-b^2*d)^(2/3)*(d*x + c))/(d*x + c)) + 3*(3*b
^3*d^3*f*x^2 - 7*(b^3*c*d^2 - a*b^2*d^3)*f*x - (28*b^3*c^2*d - 25*a*b^2*c*d^2)*f + 6*(b^3*d^3*x + 4*b^3*c*d^2
- 3*a*b^2*d^3)*e)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^2*d^5*x + b^2*c*d^4), 1/18*(12*sqrt(1/3)*((7*b^3*c^2*d^2
- 8*a*b^2*c*d^3 + a^2*b*d^4)*f*x + (7*b^3*c^3*d - 8*a*b^2*c^2*d^2 + a^2*b*c*d^3)*f - 6*(b^3*c^2*d^2 - a*b^2*c
*d^3 + (b^3*c*d^3 - a*b^2*d^4)*x)*e)*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)) + 2*(-b^2*d)^(2/3
)*((7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*f*x + (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f - 6*(b^2*c^2*d - a*b*c*
d^2 + (b^2*c*d^2 - a*b*d^3)*x)*e)*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)) - 4*(-b^2*d)^(2/3)*((7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*
d^3)*f*x + (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f - 6*(b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x)*e)*lo
g(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (-b^2*d)^(2/3)*(d*x + c))/(d*x + c)) + 3*(3*b^3*d^3*f*x^2 - 7*(b^3*c*
d^2 - a*b^2*d^3)*f*x - (28*b^3*c^2*d - 25*a*b^2*c*d^2)*f + 6*(b^3*d^3*x + 4*b^3*c*d^2 - 3*a*b^2*d^3)*e)*(b*x +
a)^(1/3)*(d*x + c)^(2/3))/(b^2*d^5*x + b^2*c*d^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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