\(\int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx\) [555]

Optimal result

Integrand size = 19, antiderivative size = 150 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=-\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{d^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 d^{4/3}}-\frac {3 \sqrt [3]{b} \log \left (1-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 d^{4/3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\frac {-\frac {6 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\sqrt [3]{b} \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 d^{4/3}} \] Input:

Integrate[(a + b*x)^(1/3)/(c + d*x)^(4/3),x]
 

Output:

((-6*d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) - 2*Sqrt[3]*b^(1/3)*ArcTan[( 
Sqrt[3]*b^(1/3)*(c + d*x)^(1/3))/(2*d^(1/3)*(a + b*x)^(1/3) + b^(1/3)*(c + 
 d*x)^(1/3))] - 2*b^(1/3)*Log[d^(1/3)*(a + b*x)^(1/3) - b^(1/3)*(c + d*x)^ 
(1/3)] + b^(1/3)*Log[d^(2/3)*(a + b*x)^(2/3) + b^(1/3)*d^(1/3)*(a + b*x)^( 
1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3)])/(2*d^(4/3))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {57, 71}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {b \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}}dx}{d}-\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 71

\(\displaystyle \frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}\right )}{d}-\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}\)

Input:

Int[(a + b*x)^(1/3)/(c + d*x)^(4/3),x]
 

Output:

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

Defintions of rubi rules used

rule 57

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

rule 71

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] && PosQ[d/b]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (x d +c \right )^{\frac {4}{3}}}d x\]

Input:

int((b*x+a)^(1/3)/(d*x+c)^(4/3),x)
 

Output:

int((b*x+a)^(1/3)/(d*x+c)^(4/3),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (110) = 220\).

Time = 0.09 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=-\frac {2 \, \sqrt {3} {\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} d \left (-\frac {b}{d}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) + {\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 2 \, {\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) + 6 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (d^{2} x + c d\right )}} \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(4/3),x, algorithm="fricas")
 

Output:

-1/2*(2*sqrt(3)*(d*x + c)*(-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)) 
+ (d*x + c)*(-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*(d*x + c)*(-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)) + 6*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(d^2*x + c*d 
)
 

Sympy [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int \frac {\sqrt [3]{a + b x}}{\left (c + d x\right )^{\frac {4}{3}}}\, dx \] Input:

integrate((b*x+a)**(1/3)/(d*x+c)**(4/3),x)
 

Output:

Integral((a + b*x)**(1/3)/(c + d*x)**(4/3), x)
 

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(4/3),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/3)/(d*x + c)^(4/3), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate((b*x+a)^(1/3)/(d*x+c)^(4/3),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/3)/(d*x + c)^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \] Input:

int((a + b*x)^(1/3)/(c + d*x)^(4/3),x)
 

Output:

int((a + b*x)^(1/3)/(c + d*x)^(4/3), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} c +\left (d x +c \right )^{\frac {1}{3}} d x}d x \] Input:

int((b*x+a)^(1/3)/(d*x+c)^(4/3),x)
 

Output:

int((a + b*x)**(1/3)/((c + d*x)**(1/3)*c + (c + d*x)**(1/3)*d*x),x)