\(\int \frac {(e x)^{9/2}}{(c+d x) (a x+b x^2)^{3/2}} \, dx\) [186]

Optimal result

Integrand size = 28, antiderivative size = 185 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 a^3 e^4 \sqrt {e x}}{b^3 (b c-a d) \sqrt {a x+b x^2}}-\frac {2 (b c+2 a d) e^5 \sqrt {a x+b x^2}}{b^3 d^2 \sqrt {e x}}+\frac {2 e^6 \left (a x+b x^2\right )^{3/2}}{3 b^3 d (e x)^{3/2}}+\frac {2 c^3 e^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {e} \sqrt {a x+b x^2}}{\sqrt {b c-a d} \sqrt {e x}}\right )}{d^{5/2} (b c-a d)^{3/2}} \] Output:

2*a^3*e^4*(e*x)^(1/2)/b^3/(-a*d+b*c)/(b*x^2+a*x)^(1/2)-2*(2*a*d+b*c)*e^5*( 
b*x^2+a*x)^(1/2)/b^3/d^2/(e*x)^(1/2)+2/3*e^6*(b*x^2+a*x)^(3/2)/b^3/d/(e*x) 
^(3/2)+2*c^3*e^(9/2)*arctan(d^(1/2)*e^(1/2)*(b*x^2+a*x)^(1/2)/(-a*d+b*c)^( 
1/2)/(e*x)^(1/2))/d^(5/2)/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.91 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 e^4 \sqrt {e x} \left (-\sqrt {d} \sqrt {b c-a d} \left (-8 a^3 d^2+2 a^2 b d (c-2 d x)+b^3 c x (3 c-d x)+a b^2 \left (3 c^2+c d x+d^2 x^2\right )\right )+3 b^3 c^3 \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{3 b^3 d^{5/2} (b c-a d)^{3/2} \sqrt {x (a+b x)}} \] Input:

Integrate[(e*x)^(9/2)/((c + d*x)*(a*x + b*x^2)^(3/2)),x]
 

Output:

(2*e^4*Sqrt[e*x]*(-(Sqrt[d]*Sqrt[b*c - a*d]*(-8*a^3*d^2 + 2*a^2*b*d*(c - 2 
*d*x) + b^3*c*x*(3*c - d*x) + a*b^2*(3*c^2 + c*d*x + d^2*x^2))) + 3*b^3*c^ 
3*Sqrt[a + b*x]*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]))/(3*b^3*d 
^(5/2)*(b*c - a*d)^(3/2)*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1261, 98, 2009}

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.

\(\displaystyle \int \frac {(e x)^{9/2}}{\left (a x+b x^2\right )^{3/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 1261

\(\displaystyle \frac {(e x)^{9/2} (a+b x)^{3/2} \int \frac {x^3}{(a+b x)^{3/2} (c+d x)}dx}{x^3 \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 98

\(\displaystyle \frac {(e x)^{9/2} (a+b x)^{3/2} \int \left (-\frac {a^3}{b^2 (b c-a d) (a+b x)^{3/2}}+\frac {-b c-a d}{b^2 d^2 \sqrt {a+b x}}+\frac {x}{b d \sqrt {a+b x}}-\frac {c^3}{d^2 (a d-b c) \sqrt {a+b x} (c+d x)}\right )dx}{x^3 \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e x)^{9/2} (a+b x)^{3/2} \left (\frac {2 a^3}{b^3 \sqrt {a+b x} (b c-a d)}+\frac {2 c^3 \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{5/2} (b c-a d)^{3/2}}-\frac {2 \sqrt {a+b x} (a d+b c)}{b^3 d^2}-\frac {2 a \sqrt {a+b x}}{b^3 d}+\frac {2 (a+b x)^{3/2}}{3 b^3 d}\right )}{x^3 \left (a x+b x^2\right )^{3/2}}\)

Input:

Int[(e*x)^(9/2)/((c + d*x)*(a*x + b*x^2)^(3/2)),x]
 

Output:

((e*x)^(9/2)*(a + b*x)^(3/2)*((2*a^3)/(b^3*(b*c - a*d)*Sqrt[a + b*x]) - (2 
*a*Sqrt[a + b*x])/(b^3*d) - (2*(b*c + a*d)*Sqrt[a + b*x])/(b^3*d^2) + (2*( 
a + b*x)^(3/2))/(3*b^3*d) + (2*c^3*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c 
 - a*d]])/(d^(5/2)*(b*c - a*d)^(3/2))))/(x^3*(a*x + b*x^2)^(3/2))
 

Defintions of rubi rules used

rule 98

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x 
_)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( 
e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] 
&& IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
 

rule 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {2 \left (-b d x +5 a d +3 b c \right ) \left (b x +a \right ) e^{5} x}{3 b^{3} d^{2} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}}+\frac {2 \left (-\frac {a^{3} d^{2}}{\left (a d -b c \right ) \sqrt {b e x +a e}}+\frac {b^{3} c^{3} \operatorname {arctanh}\left (\frac {d \sqrt {b e x +a e}}{\sqrt {e \left (a d -b c \right ) d}}\right )}{\left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\right ) e^{5} \sqrt {\left (b x +a \right ) e}\, x}{b^{3} d^{2} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}}\)	\(167\)
default	\(\frac {2 e^{4} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}\, \left (3 \sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {d \sqrt {\left (b x +a \right ) e}}{\sqrt {e \left (a d -b c \right ) d}}\right ) b^{3} c^{3}+\sqrt {e \left (a d -b c \right ) d}\, a \,b^{2} d^{2} x^{2}-\sqrt {e \left (a d -b c \right ) d}\, b^{3} c d \,x^{2}-4 \sqrt {e \left (a d -b c \right ) d}\, a^{2} b \,d^{2} x +\sqrt {e \left (a d -b c \right ) d}\, a \,b^{2} c d x +3 \sqrt {e \left (a d -b c \right ) d}\, b^{3} c^{2} x -8 \sqrt {e \left (a d -b c \right ) d}\, a^{3} d^{2}+2 \sqrt {e \left (a d -b c \right ) d}\, a^{2} b c d +3 \sqrt {e \left (a d -b c \right ) d}\, a \,b^{2} c^{2}\right )}{3 x \left (b x +a \right ) b^{3} d^{2} \left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\)	\(279\)

Input:

int((e*x)^(9/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*(-b*d*x+5*a*d+3*b*c)*(b*x+a)/b^3/d^2*e^5/(e*x)^(1/2)/(x*(b*x+a))^(1/2 
)*x+2/b^3/d^2*(-a^3*d^2/(a*d-b*c)/(b*e*x+a*e)^(1/2)+b^3*c^3/(a*d-b*c)/(e*( 
a*d-b*c)*d)^(1/2)*arctanh(d*(b*e*x+a*e)^(1/2)/(e*(a*d-b*c)*d)^(1/2)))*e^5* 
((b*x+a)*e)^(1/2)/(e*x)^(1/2)/(x*(b*x+a))^(1/2)*x
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.91 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (b^{4} c^{3} e^{4} x^{2} + a b^{3} c^{3} e^{4} x\right )} \sqrt {-\frac {e}{b c d - a d^{2}}} \log \left (-\frac {b d e x^{2} - {\left (b c - 2 \, a d\right )} e x - 2 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a x} \sqrt {e x} \sqrt {-\frac {e}{b c d - a d^{2}}}}{d x^{2} + c x}\right ) - 2 \, {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} e^{4} x^{2} - {\left (3 \, b^{3} c^{2} + a b^{2} c d - 4 \, a^{2} b d^{2}\right )} e^{4} x - {\left (3 \, a b^{2} c^{2} + 2 \, a^{2} b c d - 8 \, a^{3} d^{2}\right )} e^{4}\right )} \sqrt {b x^{2} + a x} \sqrt {e x}}{3 \, {\left ({\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2} + {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left (b^{4} c^{3} e^{4} x^{2} + a b^{3} c^{3} e^{4} x\right )} \sqrt {\frac {e}{b c d - a d^{2}}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {e x} \sqrt {\frac {e}{b c d - a d^{2}}}}{b e x^{2} + a e x}\right ) + {\left ({\left (b^{3} c d - a b^{2} d^{2}\right )} e^{4} x^{2} - {\left (3 \, b^{3} c^{2} + a b^{2} c d - 4 \, a^{2} b d^{2}\right )} e^{4} x - {\left (3 \, a b^{2} c^{2} + 2 \, a^{2} b c d - 8 \, a^{3} d^{2}\right )} e^{4}\right )} \sqrt {b x^{2} + a x} \sqrt {e x}\right )}}{3 \, {\left ({\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{2} + {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )} x\right )}}\right ] \] Input:

integrate((e*x)^(9/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
 

Output:

[-1/3*(3*(b^4*c^3*e^4*x^2 + a*b^3*c^3*e^4*x)*sqrt(-e/(b*c*d - a*d^2))*log( 
-(b*d*e*x^2 - (b*c - 2*a*d)*e*x - 2*(b*c*d - a*d^2)*sqrt(b*x^2 + a*x)*sqrt 
(e*x)*sqrt(-e/(b*c*d - a*d^2)))/(d*x^2 + c*x)) - 2*((b^3*c*d - a*b^2*d^2)* 
e^4*x^2 - (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*e^4*x - (3*a*b^2*c^2 + 2*a 
^2*b*c*d - 8*a^3*d^2)*e^4)*sqrt(b*x^2 + a*x)*sqrt(e*x))/((b^5*c*d^2 - a*b^ 
4*d^3)*x^2 + (a*b^4*c*d^2 - a^2*b^3*d^3)*x), 2/3*(3*(b^4*c^3*e^4*x^2 + a*b 
^3*c^3*e^4*x)*sqrt(e/(b*c*d - a*d^2))*arctan(-sqrt(b*x^2 + a*x)*(b*c - a*d 
)*sqrt(e*x)*sqrt(e/(b*c*d - a*d^2))/(b*e*x^2 + a*e*x)) + ((b^3*c*d - a*b^2 
*d^2)*e^4*x^2 - (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*e^4*x - (3*a*b^2*c^2 
 + 2*a^2*b*c*d - 8*a^3*d^2)*e^4)*sqrt(b*x^2 + a*x)*sqrt(e*x))/((b^5*c*d^2 
- a*b^4*d^3)*x^2 + (a*b^4*c*d^2 - a^2*b^3*d^3)*x)]
 

Sympy [F]

\[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\left (e x\right )^{\frac {9}{2}}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:

integrate((e*x)**(9/2)/(d*x+c)/(b*x**2+a*x)**(3/2),x)
 

Output:

Integral((e*x)**(9/2)/((x*(a + b*x))**(3/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:

integrate((e*x)^(9/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
 

Output:

2/3*((b^3*d*e^(9/2)*x^2 - a*b^2*d*e^(9/2)*x - 2*a^2*b*d*e^(9/2))*x^3 - 2*( 
a^2*b*c*e^(9/2) + a^3*d*e^(9/2) + (b^3*c*e^(9/2) + a*b^2*d*e^(9/2))*x^2 + 
2*(a*b^2*c*e^(9/2) + a^2*b*d*e^(9/2))*x)*x^2)/((b^4*d^2*x^3 + a*b^3*c*d*x 
+ (b^4*c*d + a*b^3*d^2)*x^2)*sqrt(b*x + a)) + integrate(2/3*((4*a^3*b*c*d* 
e^(9/2) + (3*b^4*c^2*e^(9/2) + 4*a*b^3*c*d*e^(9/2) + 3*a^2*b^2*d^2*e^(9/2) 
)*x^2 + (3*a*b^3*c^2*e^(9/2) + 8*a^2*b^2*c*d*e^(9/2) + 3*a^3*b*d^2*e^(9/2) 
)*x)*x^3 + (2*a^3*b*c^2*e^(9/2) + 2*a^4*c*d*e^(9/2) + 5*(a*b^3*c^2*e^(9/2) 
 + a^2*b^2*c*d*e^(9/2))*x^2 + 7*(a^2*b^2*c^2*e^(9/2) + a^3*b*c*d*e^(9/2))* 
x)*x^2)/((b^5*d^3*x^6 + a^2*b^3*c^2*d*x^2 + 2*(b^5*c*d^2 + a*b^4*d^3)*x^5 
+ (b^5*c^2*d + 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^4 + 2*(a*b^4*c^2*d + a^2*b^3 
*c*d^2)*x^3)*sqrt(b*x + a)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (155) = 310\).

Time = 0.25 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.04 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2}{3} \, {\left (\frac {3 \, c^{3} e^{2} \arctan \left (\frac {\sqrt {b e x + a e} d}{\sqrt {b c d e - a d^{2} e}}\right )}{{\left (b c d^{2} {\left | e \right |} - a d^{3} {\left | e \right |}\right )} \sqrt {b c d e - a d^{2} e}} + \frac {3 \, a^{3} e^{2}}{{\left (b^{4} c {\left | e \right |} - a b^{3} d {\left | e \right |}\right )} \sqrt {b e x + a e}} - \frac {3 \, \sqrt {b e x + a e} b^{7} c d e^{3} + 6 \, \sqrt {b e x + a e} a b^{6} d^{2} e^{3} - {\left (b e x + a e\right )}^{\frac {3}{2}} b^{6} d^{2} e^{2}}{b^{9} d^{3} e^{2} {\left | e \right |}}\right )} e^{4} - \frac {2 \, {\left (3 \, \sqrt {a e} b^{3} c^{3} e^{6} \arctan \left (\frac {\sqrt {a e} d}{\sqrt {b c d e - a d^{2} e}}\right ) - 3 \, \sqrt {b c d e - a d^{2} e} a b^{2} c^{2} e^{6} - 2 \, \sqrt {b c d e - a d^{2} e} a^{2} b c d e^{6} + 8 \, \sqrt {b c d e - a d^{2} e} a^{3} d^{2} e^{6}\right )}}{3 \, {\left (\sqrt {b c d e - a d^{2} e} \sqrt {a e} b^{4} c d^{2} {\left | e \right |} - \sqrt {b c d e - a d^{2} e} \sqrt {a e} a b^{3} d^{3} {\left | e \right |}\right )}} \] Input:

integrate((e*x)^(9/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
 

Output:

2/3*(3*c^3*e^2*arctan(sqrt(b*e*x + a*e)*d/sqrt(b*c*d*e - a*d^2*e))/((b*c*d 
^2*abs(e) - a*d^3*abs(e))*sqrt(b*c*d*e - a*d^2*e)) + 3*a^3*e^2/((b^4*c*abs 
(e) - a*b^3*d*abs(e))*sqrt(b*e*x + a*e)) - (3*sqrt(b*e*x + a*e)*b^7*c*d*e^ 
3 + 6*sqrt(b*e*x + a*e)*a*b^6*d^2*e^3 - (b*e*x + a*e)^(3/2)*b^6*d^2*e^2)/( 
b^9*d^3*e^2*abs(e)))*e^4 - 2/3*(3*sqrt(a*e)*b^3*c^3*e^6*arctan(sqrt(a*e)*d 
/sqrt(b*c*d*e - a*d^2*e)) - 3*sqrt(b*c*d*e - a*d^2*e)*a*b^2*c^2*e^6 - 2*sq 
rt(b*c*d*e - a*d^2*e)*a^2*b*c*d*e^6 + 8*sqrt(b*c*d*e - a*d^2*e)*a^3*d^2*e^ 
6)/(sqrt(b*c*d*e - a*d^2*e)*sqrt(a*e)*b^4*c*d^2*abs(e) - sqrt(b*c*d*e - a* 
d^2*e)*sqrt(a*e)*a*b^3*d^3*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{{\left (b\,x^2+a\,x\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:

int((e*x)^(9/2)/((a*x + b*x^2)^(3/2)*(c + d*x)),x)
 

Output:

int((e*x)^(9/2)/((a*x + b*x^2)^(3/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.19 \[ \int \frac {(e x)^{9/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {e}\, e^{4} \left (3 \sqrt {d}\, \sqrt {b x +a}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) b^{3} c^{3}-8 a^{4} d^{4}+10 a^{3} b c \,d^{3}-4 a^{3} b \,d^{4} x +a^{2} b^{2} c^{2} d^{2}+5 a^{2} b^{2} c \,d^{3} x +a^{2} b^{2} d^{4} x^{2}-3 a \,b^{3} c^{3} d +2 a \,b^{3} c^{2} d^{2} x -2 a \,b^{3} c \,d^{3} x^{2}-3 b^{4} c^{3} d x +b^{4} c^{2} d^{2} x^{2}\right )}{3 \sqrt {b x +a}\, b^{3} d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \] Input:

int((e*x)^(9/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x)
 

Output:

(2*sqrt(e)*e**4*(3*sqrt(d)*sqrt(a + b*x)*sqrt( - a*d + b*c)*atan((sqrt(a + 
 b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*b**3*c**3 - 8*a**4*d**4 + 10*a**3*b 
*c*d**3 - 4*a**3*b*d**4*x + a**2*b**2*c**2*d**2 + 5*a**2*b**2*c*d**3*x + a 
**2*b**2*d**4*x**2 - 3*a*b**3*c**3*d + 2*a*b**3*c**2*d**2*x - 2*a*b**3*c*d 
**3*x**2 - 3*b**4*c**3*d*x + b**4*c**2*d**2*x**2))/(3*sqrt(a + b*x)*b**3*d 
**3*(a**2*d**2 - 2*a*b*c*d + b**2*c**2))