\(\int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) [590]

Optimal result

Integrand size = 19, antiderivative size = 85 \[ \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )\right )}{b^{3/4} \sqrt [4]{d}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.44, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {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.

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(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))))/b
 

Defintions of rubi rules used

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]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.61 \[ \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + i \, \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, b d x + i \, b c\right )} \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - i \, \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, b d x - i \, b c\right )} \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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