3.3038 \(\int \frac{(a+b x)^{4/3} (e+f x)}{(c+d x)^{4/3}} \, dx\)
Optimal. Leaf size=328 \[ \frac{(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac{(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac{2 (b c-a d) (a d f-7 b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3} d^{10/3}}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}+\frac{3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \]
[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))
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Rubi [A] time = 0.242507, antiderivative size = 328, normalized size of antiderivative = 1.,
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 =
{78, 50, 59} \[ \frac{(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac{(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac{2 (b c-a d) (a d f-7 b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3} d^{10/3}}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}+\frac{3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \]
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 78
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] || LtQ
[p, n]))))
Rule 50
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 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]
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*}
Mathematica [C] time = 0.140826, size = 114, normalized size = 0.35 \[ \frac{3 (a+b x)^{7/3} \left (\sqrt [3]{\frac{b (c+d x)}{b c-a d}} (a d f-7 b c f+6 b d e) \, _2F_1\left (\frac{1}{3},\frac{7}{3};\frac{10}{3};\frac{d (a+b x)}{a d-b c}\right )+b (7 c f-7 d e)\right )}{7 b d \sqrt [3]{c+d x} (a d-b c)} \]
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 = 0.052, size = 0, normalized size = 0. \begin{align*} \int{(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)*(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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}{\left (f x + e\right )}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
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] time = 2.70699, size = 3090, normalized size = 9.42 \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)*(f*x+e)/(d*x+c)^(4/3),x, algorithm="fricas")
[Out]
[-1/18*(6*sqrt(1/3)*(6*(b^3*c^2*d^2 - a*b^2*c*d^3)*e - (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*d^3 - a*b^2*d^4)*e - (7*b^3*c^2*d^2 - 8*a*b^2*c*d^3 + a^2*b*d^4)*f)*x)*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)*(6*(b^2*c^2*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f + (6*(b^
2*c*d^2 - a*b*d^3)*e - (7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*f)*x)*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)*(
6*(b^2*c^2*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f + (6*(b^2*c*d^2 - a*b*d^3)*e - (7*b^2*c^
2*d - 8*a*b*c*d^2 + a^2*d^3)*f)*x)*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 + 6*(4*b^3*c*d^2 - 3*a*b^2*d^3)*e - (28*b^3*c^2*d - 25*a*b^2*c*d^2)*f + (6*b^3*d^3*e
- 7*(b^3*c*d^2 - a*b^2*d^3)*f)*x)*(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)*(6*(b^3*c^2*d^2 - a*b^2*c*d^3)*e - (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*d^3 - a*b^2*
d^4)*e - (7*b^3*c^2*d^2 - 8*a*b^2*c*d^3 + a^2*b*d^4)*f)*x)*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)*(6*(b^2*c^2*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f + (6*(b^2*c*d^2
- a*b*d^3)*e - (7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*f)*x)*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)*(6*(b^2*
c^2*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f + (6*(b^2*c*d^2 - a*b*d^3)*e - (7*b^2*c^2*d - 8
*a*b*c*d^2 + a^2*d^3)*f)*x)*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 + 6*(4*b^3*c*d^2 - 3*a*b^2*d^3)*e - (28*b^3*c^2*d - 25*a*b^2*c*d^2)*f + (6*b^3*d^3*e - 7*(b
^3*c*d^2 - a*b^2*d^3)*f)*x)*(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., size = 0, normalized size = 0. \begin{align*} \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{align*}
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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}{\left (f x + e\right )}}{{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
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)