\(\int \frac {(e x)^{7/2}}{(c+d x) (a x+b x^2)^{3/2}} \, dx\) [187]

Optimal result

Integrand size = 28, antiderivative size = 144 \[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 a^2 e^3 \sqrt {e x}}{b^2 (b c-a d) \sqrt {a x+b x^2}}+\frac {2 e^4 \sqrt {a x+b x^2}}{b^2 d \sqrt {e x}}-\frac {2 c^2 e^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {e} \sqrt {a x+b x^2}}{\sqrt {b c-a d} \sqrt {e x}}\right )}{d^{3/2} (b c-a d)^{3/2}} \] Output:

-2*a^2*e^3*(e*x)^(1/2)/b^2/(-a*d+b*c)/(b*x^2+a*x)^(1/2)+2*e^4*(b*x^2+a*x)^ 
(1/2)/b^2/d/(e*x)^(1/2)-2*c^2*e^(7/2)*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^(3/2)/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 e^3 \sqrt {e x} \left (\sqrt {d} \sqrt {b c-a d} \left (-2 a^2 d+b^2 c x+a b (c-d x)\right )-b^2 c^2 \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{b^2 d^{3/2} (b c-a d)^{3/2} \sqrt {x (a+b x)}} \] Input:

Integrate[(e*x)^(7/2)/((c + d*x)*(a*x + b*x^2)^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89, 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.

\(\displaystyle \int \frac {(e x)^{7/2}}{\left (a x+b x^2\right )^{3/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 1261

\(\displaystyle \frac {(e x)^{7/2} (a+b x)^{3/2} \int \frac {x^2}{(a+b x)^{3/2} (c+d x)}dx}{x^2 \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 98

\(\displaystyle \frac {(e x)^{7/2} (a+b x)^{3/2} \int \left (\frac {a^2}{b (b c-a d) (a+b x)^{3/2}}+\frac {1}{b d \sqrt {a+b x}}+\frac {c^2}{d (a d-b c) \sqrt {a+b x} (c+d x)}\right )dx}{x^2 \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e x)^{7/2} (a+b x)^{3/2} \left (-\frac {2 a^2}{b^2 \sqrt {a+b x} (b c-a d)}-\frac {2 c^2 \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} (b c-a d)^{3/2}}+\frac {2 \sqrt {a+b x}}{b^2 d}\right )}{x^2 \left (a x+b x^2\right )^{3/2}}\)

Input:

Int[(e*x)^(7/2)/((c + d*x)*(a*x + b*x^2)^(3/2)),x]
 

Output:

((e*x)^(7/2)*(a + b*x)^(3/2)*((-2*a^2)/(b^2*(b*c - a*d)*Sqrt[a + b*x]) + ( 
2*Sqrt[a + b*x])/(b^2*d) - (2*c^2*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c 
- a*d]])/(d^(3/2)*(b*c - a*d)^(3/2))))/(x^2*(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 = 0.54 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.05

method	result	size
risch	\(\frac {2 \left (b x +a \right ) e^{4} x}{d \,b^{2} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}}-\frac {2 \left (-\frac {a^{2} d}{\left (a d -b c \right ) \sqrt {b e x +a e}}+\frac {b^{2} c^{2} \operatorname {arctanh}\left (\frac {d \sqrt {b e x +a e}}{\sqrt {e \left (a d -b c \right ) d}}\right )}{\left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\right ) e^{4} \sqrt {\left (b x +a \right ) e}\, x}{b^{2} d \sqrt {e x}\, \sqrt {x \left (b x +a \right )}}\)	\(151\)
default	\(-\frac {2 e^{3} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}\, \left (\sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {d \sqrt {\left (b x +a \right ) e}}{\sqrt {e \left (a d -b c \right ) d}}\right ) b^{2} c^{2}-\sqrt {e \left (a d -b c \right ) d}\, a b d x +\sqrt {e \left (a d -b c \right ) d}\, b^{2} c x -2 \sqrt {e \left (a d -b c \right ) d}\, a^{2} d +\sqrt {e \left (a d -b c \right ) d}\, a b c \right )}{x \left (b x +a \right ) b^{2} d \left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\)	\(175\)

Input:

int((e*x)^(7/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/d/b^2*(b*x+a)*e^4/(e*x)^(1/2)/(x*(b*x+a))^(1/2)*x-2/b^2/d*(-a^2*d/(a*d-b 
*c)/(b*e*x+a*e)^(1/2)+b^2*c^2/(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^4*((b*x+a)*e)^(1/2)/(e*x)^(1/2)/ 
(x*(b*x+a))^(1/2)*x
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.93 \[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (b^{3} c^{2} e^{3} x^{2} + a b^{2} c^{2} e^{3} x\right )} \sqrt {-\frac {e}{b c d - a d^{2}}} \log \left (-\frac {b d e x^{2} - {\left (b c - 2 \, a d\right )} e x + 2 \, {\left (b c d - a d^{2}\right )} \sqrt {b x^{2} + a x} \sqrt {e x} \sqrt {-\frac {e}{b c d - a d^{2}}}}{d x^{2} + c x}\right ) - 2 \, {\left ({\left (b^{2} c - a b d\right )} e^{3} x + {\left (a b c - 2 \, a^{2} d\right )} e^{3}\right )} \sqrt {b x^{2} + a x} \sqrt {e x}}{{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{2} + {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} x}, -\frac {2 \, {\left ({\left (b^{3} c^{2} e^{3} x^{2} + a b^{2} c^{2} e^{3} x\right )} \sqrt {\frac {e}{b c d - a d^{2}}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {e x} \sqrt {\frac {e}{b c d - a d^{2}}}}{b e x^{2} + a e x}\right ) - {\left ({\left (b^{2} c - a b d\right )} e^{3} x + {\left (a b c - 2 \, a^{2} d\right )} e^{3}\right )} \sqrt {b x^{2} + a x} \sqrt {e x}\right )}}{{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{2} + {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} x}\right ] \] Input:

integrate((e*x)^(7/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
 

Output:

[-((b^3*c^2*e^3*x^2 + a*b^2*c^2*e^3*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^2*c - a*b*d)*e^3*x + (a*b 
*c - 2*a^2*d)*e^3)*sqrt(b*x^2 + a*x)*sqrt(e*x))/((b^4*c*d - a*b^3*d^2)*x^2 
 + (a*b^3*c*d - a^2*b^2*d^2)*x), -2*((b^3*c^2*e^3*x^2 + a*b^2*c^2*e^3*x)*s 
qrt(e/(b*c*d - a*d^2))*arctan(-sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(e*x)*sqr 
t(e/(b*c*d - a*d^2))/(b*e*x^2 + a*e*x)) - ((b^2*c - a*b*d)*e^3*x + (a*b*c 
- 2*a^2*d)*e^3)*sqrt(b*x^2 + a*x)*sqrt(e*x))/((b^4*c*d - a*b^3*d^2)*x^2 + 
(a*b^3*c*d - a^2*b^2*d^2)*x)]
 

Sympy [F]

\[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\left (e x\right )^{\frac {7}{2}}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:

integrate((e*x)**(7/2)/(d*x+c)/(b*x**2+a*x)**(3/2),x)
 

Output:

Integral((e*x)**(7/2)/((x*(a + b*x))**(3/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:

integrate((e*x)^(7/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (120) = 240\).

Time = 0.18 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.93 \[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=2 \, e^{3} {\left (\frac {\sqrt {a e} b^{2} c^{2} e^{2} \arctan \left (\frac {\sqrt {a e} d}{\sqrt {b c d e - a d^{2} e}}\right ) - \sqrt {b c d e - a d^{2} e} a b c e^{2} + 2 \, \sqrt {b c d e - a d^{2} e} a^{2} d e^{2}}{\sqrt {b c d e - a d^{2} e} \sqrt {a e} b^{3} c d {\left | e \right |} - \sqrt {b c d e - a d^{2} e} \sqrt {a e} a b^{2} d^{2} {\left | e \right |}} - \frac {{\left (\frac {b^{2} c^{2} e \arctan \left (\frac {\sqrt {b e x + a e} d}{\sqrt {b c d e - a d^{2} e}}\right )}{\sqrt {b c d e - a d^{2} e} {\left (b c d {\left | e \right |} - a d^{2} {\left | e \right |}\right )}} + \frac {a^{2} e}{\sqrt {b e x + a e} {\left (b c {\left | e \right |} - a d {\left | e \right |}\right )}} - \frac {\sqrt {b e x + a e}}{d {\left | e \right |}}\right )} e}{b^{2}}\right )} \] Input:

integrate((e*x)^(7/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
 

Output:

2*e^3*((sqrt(a*e)*b^2*c^2*e^2*arctan(sqrt(a*e)*d/sqrt(b*c*d*e - a*d^2*e)) 
- sqrt(b*c*d*e - a*d^2*e)*a*b*c*e^2 + 2*sqrt(b*c*d*e - a*d^2*e)*a^2*d*e^2) 
/(sqrt(b*c*d*e - a*d^2*e)*sqrt(a*e)*b^3*c*d*abs(e) - sqrt(b*c*d*e - a*d^2* 
e)*sqrt(a*e)*a*b^2*d^2*abs(e)) - (b^2*c^2*e*arctan(sqrt(b*e*x + a*e)*d/sqr 
t(b*c*d*e - a*d^2*e))/(sqrt(b*c*d*e - a*d^2*e)*(b*c*d*abs(e) - a*d^2*abs(e 
))) + a^2*e/(sqrt(b*e*x + a*e)*(b*c*abs(e) - a*d*abs(e))) - sqrt(b*e*x + a 
*e)/(d*abs(e)))*e/b^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\,x\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:

int((e*x)^(7/2)/((a*x + b*x^2)^(3/2)*(c + d*x)),x)
 

Output:

int((e*x)^(7/2)/((a*x + b*x^2)^(3/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04 \[ \int \frac {(e x)^{7/2}}{(c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {e}\, e^{3} \left (-\sqrt {d}\, \sqrt {b x +a}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) b^{2} c^{2}+2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+a^{2} b \,d^{3} x +a \,b^{2} c^{2} d -2 a \,b^{2} c \,d^{2} x +b^{3} c^{2} d x \right )}{\sqrt {b x +a}\, b^{2} d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \] Input:

int((e*x)^(7/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x)
 

Output:

(2*sqrt(e)*e**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**2*c**2 + 2*a**3*d**3 - 3*a**2*b 
*c*d**2 + a**2*b*d**3*x + a*b**2*c**2*d - 2*a*b**2*c*d**2*x + b**3*c**2*d* 
x))/(sqrt(a + b*x)*b**2*d**2*(a**2*d**2 - 2*a*b*c*d + b**2*c**2))