\(\int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx\) [614]

Optimal result

Integrand size = 19, antiderivative size = 108 \[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}+\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{d^{5/4}} \] Output:

-4*(b*x+a)^(1/4)/d/(d*x+c)^(1/4)+2*b^(1/4)*arctan(d^(1/4)*(b*x+a)^(1/4)/b^ 
(1/4)/(d*x+c)^(1/4))/d^(5/4)+2*b^(1/4)*arctanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/ 
4)/(d*x+c)^(1/4))/d^(5/4)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}-\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{d^{5/4}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{d^{5/4}} \] Input:

Integrate[(a + b*x)^(1/4)/(c + d*x)^(5/4),x]
 

Output:

(-4*(a + b*x)^(1/4))/(d*(c + d*x)^(1/4)) - (2*b^(1/4)*ArcTan[(b^(1/4)*(c + 
 d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))])/d^(5/4) + (2*b^(1/4)*ArcTanh[(b^( 
1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))])/d^(5/4)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {57, 73, 770, 756, 218, 221}

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.

\(\displaystyle \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {b \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{d}-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 \int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [4]{a+b x}}{d}-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {4 \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{d}-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {4 \left (\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}+\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{d}-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {4 \left (\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}\right )}{d}-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {4 \left (\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}\right )}{d}-\frac {4 \sqrt [4]{a+b x}}{d \sqrt [4]{c+d x}}\)

Input:

Int[(a + b*x)^(1/4)/(c + d*x)^(5/4),x]
 

Output:

(-4*(a + b*x)^(1/4))/(d*(c + d*x)^(1/4)) + (4*((b^(1/4)*ArcTan[(d^(1/4)*(a 
 + b*x)^(1/4))/(b^(1/4)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))])/(2*d^(1/4 
)) + (b^(1/4)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c - (a*d)/b + (d 
*(a + b*x))/b)^(1/4))])/(2*d^(1/4))))/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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (x d +c \right )^{\frac {5}{4}}}d x\]

Input:

int((b*x+a)^(1/4)/(d*x+c)^(5/4),x)
 

Output:

int((b*x+a)^(1/4)/(d*x+c)^(5/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - {\left (d^{2} x + c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d^{2} x + c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + {\left (i \, d^{2} x + i \, c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, d^{2} x + i \, c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + {\left (-i \, d^{2} x - i \, c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, d^{2} x - i \, c d\right )} \left (\frac {b}{d^{5}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - 4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d^{2} x + c d} \] Input:

integrate((b*x+a)^(1/4)/(d*x+c)^(5/4),x, algorithm="fricas")
 

Output:

((d^2*x + c*d)*(b/d^5)^(1/4)*log(((d^2*x + c*d)*(b/d^5)^(1/4) + (b*x + a)^ 
(1/4)*(d*x + c)^(3/4))/(d*x + c)) - (d^2*x + c*d)*(b/d^5)^(1/4)*log(-((d^2 
*x + c*d)*(b/d^5)^(1/4) - (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + (I 
*d^2*x + I*c*d)*(b/d^5)^(1/4)*log(((I*d^2*x + I*c*d)*(b/d^5)^(1/4) + (b*x 
+ a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + (-I*d^2*x - I*c*d)*(b/d^5)^(1/4)* 
log(((-I*d^2*x - I*c*d)*(b/d^5)^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/( 
d*x + c)) - 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(d^2*x + c*d)
 

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=\int \frac {\sqrt [4]{a + b x}}{\left (c + d x\right )^{\frac {5}{4}}}\, dx \] Input:

integrate((b*x+a)**(1/4)/(d*x+c)**(5/4),x)
 

Output:

Integral((a + b*x)**(1/4)/(c + d*x)**(5/4), x)
 

Maxima [F]

\[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((b*x+a)^(1/4)/(d*x+c)^(5/4),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/4)/(d*x + c)^(5/4), x)
 

Giac [F]

\[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((b*x+a)^(1/4)/(d*x+c)^(5/4),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/4)/(d*x + c)^(5/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/4}}{{\left (c+d\,x\right )}^{5/4}} \,d x \] Input:

int((a + b*x)^(1/4)/(c + d*x)^(5/4),x)
 

Output:

int((a + b*x)^(1/4)/(c + d*x)^(5/4), x)
 

Reduce [F]

\[ \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{5/4}} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}} c +\left (d x +c \right )^{\frac {1}{4}} d x}d x \] Input:

int((b*x+a)^(1/4)/(d*x+c)^(5/4),x)
 

Output:

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