\(\int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx\) [1672]

Optimal result

Integrand size = 26, antiderivative size = 236 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\frac {5 (d e-c f) \sqrt [4]{c+d x}}{(b e-a f)^2 \sqrt [4]{e+f x}}-\frac {(c+d x)^{5/4}}{(b e-a f) (a+b x) \sqrt [4]{e+f x}}-\frac {5 \sqrt [4]{b c-a d} (d e-c f) \arctan \left (\frac {\sqrt [4]{b e-a f} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d} \sqrt [4]{e+f x}}\right )}{2 (b e-a f)^{9/4}}-\frac {5 \sqrt [4]{b c-a d} (d e-c f) \text {arctanh}\left (\frac {\sqrt [4]{b e-a f} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d} \sqrt [4]{e+f x}}\right )}{2 (b e-a f)^{9/4}} \] Output:

5*(-c*f+d*e)*(d*x+c)^(1/4)/(-a*f+b*e)^2/(f*x+e)^(1/4)-(d*x+c)^(5/4)/(-a*f+ 
b*e)/(b*x+a)/(f*x+e)^(1/4)-5/2*(-a*d+b*c)^(1/4)*(-c*f+d*e)*arctan((-a*f+b* 
e)^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4)/(f*x+e)^(1/4))/(-a*f+b*e)^(9/4)-5/ 
2*(-a*d+b*c)^(1/4)*(-c*f+d*e)*arctanh((-a*f+b*e)^(1/4)*(d*x+c)^(1/4)/(-a*d 
+b*c)^(1/4)/(f*x+e)^(1/4))/(-a*f+b*e)^(9/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.52 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\frac {\sqrt [4]{c+d x} \left (4 b d e x-b c (e+5 f x)+a (5 d e-4 c f+d f x)-5 (d e-c f) (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )\right )}{(b e-a f)^2 (a+b x) \sqrt [4]{e+f x}} \] Input:

Integrate[(c + d*x)^(5/4)/((a + b*x)^2*(e + f*x)^(5/4)),x]
 

Output:

((c + d*x)^(1/4)*(4*b*d*e*x - b*c*(e + 5*f*x) + a*(5*d*e - 4*c*f + d*f*x) 
- 5*(d*e - c*f)*(a + b*x)*Hypergeometric2F1[1/4, 1, 5/4, ((b*e - a*f)*(c + 
 d*x))/((b*c - a*d)*(e + f*x))]))/((b*e - a*f)^2*(a + b*x)*(e + f*x)^(1/4) 
)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {105, 105, 104, 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.

\(\displaystyle \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

\(\displaystyle \frac {5 (d e-c f) \left (\frac {(b c-a d) \int \frac {1}{(a+b x) (c+d x)^{3/4} \sqrt [4]{e+f x}}dx}{b e-a f}+\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{e+f x} (b e-a f)}\right )}{4 (b e-a f)}-\frac {(c+d x)^{5/4}}{(a+b x) \sqrt [4]{e+f x} (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {5 (d e-c f) \left (\frac {4 (b c-a d) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt [4]{c+d x}}{\sqrt [4]{e+f x}}}{b e-a f}+\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{e+f x} (b e-a f)}\right )}{4 (b e-a f)}-\frac {(c+d x)^{5/4}}{(a+b x) \sqrt [4]{e+f x} (b e-a f)}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {5 (d e-c f) \left (\frac {4 (b c-a d) \left (-\frac {\int \frac {1}{\sqrt {b c-a d}-\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {e+f x}}}d\frac {\sqrt [4]{c+d x}}{\sqrt [4]{e+f x}}}{2 \sqrt {b c-a d}}-\frac {\int \frac {1}{\sqrt {b c-a d}+\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {e+f x}}}d\frac {\sqrt [4]{c+d x}}{\sqrt [4]{e+f x}}}{2 \sqrt {b c-a d}}\right )}{b e-a f}+\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{e+f x} (b e-a f)}\right )}{4 (b e-a f)}-\frac {(c+d x)^{5/4}}{(a+b x) \sqrt [4]{e+f x} (b e-a f)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {5 (d e-c f) \left (\frac {4 (b c-a d) \left (-\frac {\int \frac {1}{\sqrt {b c-a d}-\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {e+f x}}}d\frac {\sqrt [4]{c+d x}}{\sqrt [4]{e+f x}}}{2 \sqrt {b c-a d}}-\frac {\arctan \left (\frac {\sqrt [4]{c+d x} \sqrt [4]{b e-a f}}{\sqrt [4]{e+f x} \sqrt [4]{b c-a d}}\right )}{2 (b c-a d)^{3/4} \sqrt [4]{b e-a f}}\right )}{b e-a f}+\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{e+f x} (b e-a f)}\right )}{4 (b e-a f)}-\frac {(c+d x)^{5/4}}{(a+b x) \sqrt [4]{e+f x} (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 (d e-c f) \left (\frac {4 (b c-a d) \left (-\frac {\arctan \left (\frac {\sqrt [4]{c+d x} \sqrt [4]{b e-a f}}{\sqrt [4]{e+f x} \sqrt [4]{b c-a d}}\right )}{2 (b c-a d)^{3/4} \sqrt [4]{b e-a f}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c+d x} \sqrt [4]{b e-a f}}{\sqrt [4]{e+f x} \sqrt [4]{b c-a d}}\right )}{2 (b c-a d)^{3/4} \sqrt [4]{b e-a f}}\right )}{b e-a f}+\frac {4 \sqrt [4]{c+d x}}{\sqrt [4]{e+f x} (b e-a f)}\right )}{4 (b e-a f)}-\frac {(c+d x)^{5/4}}{(a+b x) \sqrt [4]{e+f x} (b e-a f)}\)

Input:

Int[(c + d*x)^(5/4)/((a + b*x)^2*(e + f*x)^(5/4)),x]
 

Output:

-((c + d*x)^(5/4)/((b*e - a*f)*(a + b*x)*(e + f*x)^(1/4))) + (5*(d*e - c*f 
)*((4*(c + d*x)^(1/4))/((b*e - a*f)*(e + f*x)^(1/4)) + (4*(b*c - a*d)*(-1/ 
2*ArcTan[((b*e - a*f)^(1/4)*(c + d*x)^(1/4))/((b*c - a*d)^(1/4)*(e + f*x)^ 
(1/4))]/((b*c - a*d)^(3/4)*(b*e - a*f)^(1/4)) - ArcTanh[((b*e - a*f)^(1/4) 
*(c + d*x)^(1/4))/((b*c - a*d)^(1/4)*(e + f*x)^(1/4))]/(2*(b*c - a*d)^(3/4 
)*(b*e - a*f)^(1/4))))/(b*e - a*f)))/(4*(b*e - a*f))
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

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

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]
 

Maple [F]

\[\int \frac {\left (x d +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{2} \left (f x +e \right )^{\frac {5}{4}}}d x\]

Input:

int((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x)
 

Output:

int((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 2736, normalized size of antiderivative = 11.59 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x, algorithm="fricas")
 

Output:

-1/4*(5*(a*b^2*e^3 - 2*a^2*b*e^2*f + a^3*e*f^2 + (b^3*e^2*f - 2*a*b^2*e*f^ 
2 + a^2*b*f^3)*x^2 + (b^3*e^3 - a*b^2*e^2*f - a^2*b*e*f^2 + a^3*f^3)*x)*(( 
(b*c*d^4 - a*d^5)*e^4 - 4*(b*c^2*d^3 - a*c*d^4)*e^3*f + 6*(b*c^3*d^2 - a*c 
^2*d^3)*e^2*f^2 - 4*(b*c^4*d - a*c^3*d^2)*e*f^3 + (b*c^5 - a*c^4*d)*f^4)/( 
b^9*e^9 - 9*a*b^8*e^8*f + 36*a^2*b^7*e^7*f^2 - 84*a^3*b^6*e^6*f^3 + 126*a^ 
4*b^5*e^5*f^4 - 126*a^5*b^4*e^4*f^5 + 84*a^6*b^3*e^3*f^6 - 36*a^7*b^2*e^2* 
f^7 + 9*a^8*b*e*f^8 - a^9*f^9))^(1/4)*log(-5*((d*e - c*f)*(d*x + c)^(1/4)* 
(f*x + e)^(3/4) + (b^2*e^3 - 2*a*b*e^2*f + a^2*e*f^2 + (b^2*e^2*f - 2*a*b* 
e*f^2 + a^2*f^3)*x)*(((b*c*d^4 - a*d^5)*e^4 - 4*(b*c^2*d^3 - a*c*d^4)*e^3* 
f + 6*(b*c^3*d^2 - a*c^2*d^3)*e^2*f^2 - 4*(b*c^4*d - a*c^3*d^2)*e*f^3 + (b 
*c^5 - a*c^4*d)*f^4)/(b^9*e^9 - 9*a*b^8*e^8*f + 36*a^2*b^7*e^7*f^2 - 84*a^ 
3*b^6*e^6*f^3 + 126*a^4*b^5*e^5*f^4 - 126*a^5*b^4*e^4*f^5 + 84*a^6*b^3*e^3 
*f^6 - 36*a^7*b^2*e^2*f^7 + 9*a^8*b*e*f^8 - a^9*f^9))^(1/4))/(f*x + e)) - 
5*(a*b^2*e^3 - 2*a^2*b*e^2*f + a^3*e*f^2 + (b^3*e^2*f - 2*a*b^2*e*f^2 + a^ 
2*b*f^3)*x^2 + (b^3*e^3 - a*b^2*e^2*f - a^2*b*e*f^2 + a^3*f^3)*x)*(((b*c*d 
^4 - a*d^5)*e^4 - 4*(b*c^2*d^3 - a*c*d^4)*e^3*f + 6*(b*c^3*d^2 - a*c^2*d^3 
)*e^2*f^2 - 4*(b*c^4*d - a*c^3*d^2)*e*f^3 + (b*c^5 - a*c^4*d)*f^4)/(b^9*e^ 
9 - 9*a*b^8*e^8*f + 36*a^2*b^7*e^7*f^2 - 84*a^3*b^6*e^6*f^3 + 126*a^4*b^5* 
e^5*f^4 - 126*a^5*b^4*e^4*f^5 + 84*a^6*b^3*e^3*f^6 - 36*a^7*b^2*e^2*f^7 + 
9*a^8*b*e*f^8 - a^9*f^9))^(1/4)*log(-5*((d*e - c*f)*(d*x + c)^(1/4)*(f*...
 

Sympy [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{4}}}{\left (a + b x\right )^{2} \left (e + f x\right )^{\frac {5}{4}}}\, dx \] Input:

integrate((d*x+c)**(5/4)/(b*x+a)**2/(f*x+e)**(5/4),x)
 

Output:

Integral((c + d*x)**(5/4)/((a + b*x)**2*(e + f*x)**(5/4)), x)
 

Maxima [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{2} {\left (f x + e\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(5/4)/((b*x + a)^2*(f*x + e)^(5/4)), x)
 

Giac [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{2} {\left (f x + e\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((d*x + c)^(5/4)/((b*x + a)^2*(f*x + e)^(5/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/4}}{{\left (e+f\,x\right )}^{5/4}\,{\left (a+b\,x\right )}^2} \,d x \] Input:

int((c + d*x)^(5/4)/((e + f*x)^(5/4)*(a + b*x)^2),x)
 

Output:

int((c + d*x)^(5/4)/((e + f*x)^(5/4)*(a + b*x)^2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^2 (e+f x)^{5/4}} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{2} \left (f x +e \right )^{\frac {5}{4}}}d x \] Input:

int((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x)
 

Output:

int((d*x+c)^(5/4)/(b*x+a)^2/(f*x+e)^(5/4),x)