\(\int \frac {(a+b x^2)^2}{(e x)^{7/2} (c+d x)^2} \, dx\) [353]

Optimal result

Integrand size = 24, antiderivative size = 172 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=-\frac {2 a^2}{5 c^2 e (e x)^{5/2}}+\frac {4 a^2 d}{3 c^3 e^2 (e x)^{3/2}}-\frac {2 a \left (2 b c^2+3 a d^2\right )}{c^4 e^3 \sqrt {e x}}-\frac {\left (b c^2+a d^2\right )^2 \sqrt {e x}}{c^4 d e^4 (c+d x)}+\frac {\left (b c^2-7 a d^2\right ) \left (b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{9/2} d^{3/2} e^{7/2}} \] Output:

-2/5*a^2/c^2/e/(e*x)^(5/2)+4/3*a^2*d/c^3/e^2/(e*x)^(3/2)-2*a*(3*a*d^2+2*b* 
c^2)/c^4/e^3/(e*x)^(1/2)-(a*d^2+b*c^2)^2*(e*x)^(1/2)/c^4/d/e^4/(d*x+c)+(-7 
*a*d^2+b*c^2)*(a*d^2+b*c^2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^ 
(9/2)/d^(3/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=\frac {x \left (-\frac {\sqrt {c} \sqrt {d} \left (15 b^2 c^4 x^3+30 a b c^2 d x^2 (2 c+3 d x)+a^2 d \left (6 c^3-14 c^2 d x+70 c d^2 x^2+105 d^3 x^3\right )\right )}{c+d x}+15 \left (b^2 c^4-6 a b c^2 d^2-7 a^2 d^4\right ) x^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )\right )}{15 c^{9/2} d^{3/2} (e x)^{7/2}} \] Input:

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

Output:

(x*(-((Sqrt[c]*Sqrt[d]*(15*b^2*c^4*x^3 + 30*a*b*c^2*d*x^2*(2*c + 3*d*x) + 
a^2*d*(6*c^3 - 14*c^2*d*x + 70*c*d^2*x^2 + 105*d^3*x^3)))/(c + d*x)) + 15* 
(b^2*c^4 - 6*a*b*c^2*d^2 - 7*a^2*d^4)*x^(5/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqr 
t[c]]))/(15*c^(9/2)*d^(3/2)*(e*x)^(7/2))
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {518, 1583, 25, 2333, 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.

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

\(\Big \downarrow \) 518

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

\(\Big \downarrow \) 1583

\(\displaystyle \frac {2 \left (-\frac {\int -\frac {2 a^2 c^3 d^4 e^7+d^3 \left (b^2 c^4-2 a b d^2 c^2-a^2 d^4\right ) x^3 e^7+2 a c d^4 \left (2 b c^2+a d^2\right ) x^2 e^7-2 a^2 c^2 d^5 x e^7}{e^3 x^3 (c e+d x e)}d\sqrt {e x}}{2 c^4 d^4 e^4}-\frac {\sqrt {e x} \left (a d^2+b c^2\right )^2}{2 c^4 d (c e+d e x)}\right )}{e^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {\int \frac {2 a^2 c^3 d^4 e^7+d^3 \left (b^2 c^4-2 a b d^2 c^2-a^2 d^4\right ) x^3 e^7+2 a c d^4 \left (2 b c^2+a d^2\right ) x^2 e^7-2 a^2 c^2 d^5 x e^7}{e^3 x^3 (c e+d x e)}d\sqrt {e x}}{2 c^4 d^4 e^4}-\frac {\sqrt {e x} \left (a d^2+b c^2\right )^2}{2 c^4 d (c e+d e x)}\right )}{e^3}\)

\(\Big \downarrow \) 2333

\(\displaystyle \frac {2 \left (\frac {\int \left (-\frac {4 a^2 c e^3 d^5}{x^2}+\frac {2 a \left (2 b c^2+3 a d^2\right ) e^3 d^4}{x}+\frac {2 a^2 c^2 e^3 d^4}{x^3}+\frac {\left (b c^2-7 a d^2\right ) \left (b c^2+a d^2\right ) e^4 d^3}{c e+d x e}\right )d\sqrt {e x}}{2 c^4 d^4 e^4}-\frac {\sqrt {e x} \left (a d^2+b c^2\right )^2}{2 c^4 d (c e+d e x)}\right )}{e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {-\frac {2 a^2 c^2 d^4 e^6}{5 (e x)^{5/2}}+\frac {4 a^2 c d^5 e^5}{3 (e x)^{3/2}}+\frac {d^{5/2} e^{7/2} \left (b c^2-7 a d^2\right ) \left (a d^2+b c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c}}-\frac {2 a d^4 e^4 \left (3 a d^2+2 b c^2\right )}{\sqrt {e x}}}{2 c^4 d^4 e^4}-\frac {\sqrt {e x} \left (a d^2+b c^2\right )^2}{2 c^4 d (c e+d e x)}\right )}{e^3}\)

Input:

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

Output:

(2*(-1/2*((b*c^2 + a*d^2)^2*Sqrt[e*x])/(c^4*d*(c*e + d*e*x)) + ((-2*a^2*c^ 
2*d^4*e^6)/(5*(e*x)^(5/2)) + (4*a^2*c*d^5*e^5)/(3*(e*x)^(3/2)) - (2*a*d^4* 
(2*b*c^2 + 3*a*d^2)*e^4)/Sqrt[e*x] + (d^(5/2)*(b*c^2 - 7*a*d^2)*(b*c^2 + a 
*d^2)*e^(7/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/Sqrt[c])/(2*c 
^4*d^4*e^4)))/e^3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 518

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^(2*m + 1)*(e*c + d*x^2)^ 
n*(a*e^2 + b*x^4)^p, x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1583

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Sym 
bol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^( 
2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^(2*p)*(q + 1))   Int[x^ 
m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 
1)*e^(2*p)*(q + 1)*(a + c*x^4)^p - ((c*d^2 + a*e^2)^p/(e^(m/2)*x^m))*(d + e 
*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && IL 
tQ[q, -1] && ILtQ[m/2, 0]
 

rule 2009

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

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {2 a \left (45 a \,d^{2} x^{2}+30 b \,c^{2} x^{2}-10 a d x c +3 a \,c^{2}\right )}{15 c^{4} \sqrt {e x}\, x^{2} e^{3}}-\frac {\left (2 a \,d^{2}+2 b \,c^{2}\right ) \left (\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {e x}}{2 d \left (d e x +c e \right )}+\frac {\left (7 a \,d^{2}-b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 d \sqrt {d e c}}\right )}{c^{4} e^{3}}\)	\(141\)
pseudoelliptic	\(-\frac {2 \left (\frac {35 \left (d x +c \right ) x^{2} \sqrt {e x}\, \left (a \,d^{2}+b \,c^{2}\right ) \left (a \,d^{2}-\frac {b \,c^{2}}{7}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2}+\left (\frac {35 a^{2} d^{4} x^{3}}{2}+\frac {35 a^{2} c \,d^{3} x^{2}}{3}-\frac {7 x a \,c^{2} \left (-\frac {45 b \,x^{2}}{7}+a \right ) d^{2}}{3}+a \,c^{3} \left (10 b \,x^{2}+a \right ) d +\frac {5 b^{2} c^{4} x^{3}}{2}\right ) \sqrt {d e c}\right )}{5 \sqrt {e x}\, \sqrt {d e c}\, e^{3} c^{4} x^{2} \left (d x +c \right ) d}\)	\(160\)
derivativedivides	\(\frac {-\frac {2 a \left (3 a \,d^{2}+2 b \,c^{2}\right )}{c^{4} \sqrt {e x}}-\frac {2 a^{2} e^{2}}{5 c^{2} \left (e x \right )^{\frac {5}{2}}}+\frac {4 a^{2} e d}{3 c^{3} \left (e x \right )^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {e x}}{2 d \left (d e x +c e \right )}+\frac {\left (7 a^{2} d^{4}+6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 d \sqrt {d e c}}\right )}{c^{4}}}{e^{3}}\)	\(165\)
default	\(\frac {-\frac {2 a \left (3 a \,d^{2}+2 b \,c^{2}\right )}{c^{4} \sqrt {e x}}-\frac {2 a^{2} e^{2}}{5 c^{2} \left (e x \right )^{\frac {5}{2}}}+\frac {4 a^{2} e d}{3 c^{3} \left (e x \right )^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {e x}}{2 d \left (d e x +c e \right )}+\frac {\left (7 a^{2} d^{4}+6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 d \sqrt {d e c}}\right )}{c^{4}}}{e^{3}}\)	\(165\)

Input:

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

Output:

-2/15*a*(45*a*d^2*x^2+30*b*c^2*x^2-10*a*c*d*x+3*a*c^2)/c^4/(e*x)^(1/2)/x^2 
/e^3-1/c^4*(2*a*d^2+2*b*c^2)*(1/2*(a*d^2+b*c^2)/d*(e*x)^(1/2)/(d*e*x+c*e)+ 
1/2*(7*a*d^2-b*c^2)/d/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))/e 
^3
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.58 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=\left [\frac {15 \, {\left ({\left (b^{2} c^{4} d - 6 \, a b c^{2} d^{3} - 7 \, a^{2} d^{5}\right )} x^{4} + {\left (b^{2} c^{5} - 6 \, a b c^{3} d^{2} - 7 \, a^{2} c d^{4}\right )} x^{3}\right )} \sqrt {-c d e} \log \left (\frac {d e x - c e + 2 \, \sqrt {-c d e} \sqrt {e x}}{d x + c}\right ) + 2 \, {\left (14 \, a^{2} c^{3} d^{3} x - 6 \, a^{2} c^{4} d^{2} - 15 \, {\left (b^{2} c^{5} d + 6 \, a b c^{3} d^{3} + 7 \, a^{2} c d^{5}\right )} x^{3} - 10 \, {\left (6 \, a b c^{4} d^{2} + 7 \, a^{2} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {e x}}{30 \, {\left (c^{5} d^{3} e^{4} x^{4} + c^{6} d^{2} e^{4} x^{3}\right )}}, -\frac {15 \, {\left ({\left (b^{2} c^{4} d - 6 \, a b c^{2} d^{3} - 7 \, a^{2} d^{5}\right )} x^{4} + {\left (b^{2} c^{5} - 6 \, a b c^{3} d^{2} - 7 \, a^{2} c d^{4}\right )} x^{3}\right )} \sqrt {c d e} \arctan \left (\frac {\sqrt {c d e} \sqrt {e x}}{d e x}\right ) - {\left (14 \, a^{2} c^{3} d^{3} x - 6 \, a^{2} c^{4} d^{2} - 15 \, {\left (b^{2} c^{5} d + 6 \, a b c^{3} d^{3} + 7 \, a^{2} c d^{5}\right )} x^{3} - 10 \, {\left (6 \, a b c^{4} d^{2} + 7 \, a^{2} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {e x}}{15 \, {\left (c^{5} d^{3} e^{4} x^{4} + c^{6} d^{2} e^{4} x^{3}\right )}}\right ] \] Input:

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

Output:

[1/30*(15*((b^2*c^4*d - 6*a*b*c^2*d^3 - 7*a^2*d^5)*x^4 + (b^2*c^5 - 6*a*b* 
c^3*d^2 - 7*a^2*c*d^4)*x^3)*sqrt(-c*d*e)*log((d*e*x - c*e + 2*sqrt(-c*d*e) 
*sqrt(e*x))/(d*x + c)) + 2*(14*a^2*c^3*d^3*x - 6*a^2*c^4*d^2 - 15*(b^2*c^5 
*d + 6*a*b*c^3*d^3 + 7*a^2*c*d^5)*x^3 - 10*(6*a*b*c^4*d^2 + 7*a^2*c^2*d^4) 
*x^2)*sqrt(e*x))/(c^5*d^3*e^4*x^4 + c^6*d^2*e^4*x^3), -1/15*(15*((b^2*c^4* 
d - 6*a*b*c^2*d^3 - 7*a^2*d^5)*x^4 + (b^2*c^5 - 6*a*b*c^3*d^2 - 7*a^2*c*d^ 
4)*x^3)*sqrt(c*d*e)*arctan(sqrt(c*d*e)*sqrt(e*x)/(d*e*x)) - (14*a^2*c^3*d^ 
3*x - 6*a^2*c^4*d^2 - 15*(b^2*c^5*d + 6*a*b*c^3*d^3 + 7*a^2*c*d^5)*x^3 - 1 
0*(6*a*b*c^4*d^2 + 7*a^2*c^2*d^4)*x^2)*sqrt(e*x))/(c^5*d^3*e^4*x^4 + c^6*d 
^2*e^4*x^3)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=\frac {{\left (b^{2} c^{4} - 6 \, a b c^{2} d^{2} - 7 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{\sqrt {c d e} c^{4} d e^{3}} - \frac {\sqrt {e x} b^{2} c^{4} + 2 \, \sqrt {e x} a b c^{2} d^{2} + \sqrt {e x} a^{2} d^{4}}{{\left (d e x + c e\right )} c^{4} d e^{3}} - \frac {2 \, {\left (30 \, a b c^{2} e^{2} x^{2} + 45 \, a^{2} d^{2} e^{2} x^{2} - 10 \, a^{2} c d e^{2} x + 3 \, a^{2} c^{2} e^{2}\right )}}{15 \, \sqrt {e x} c^{4} e^{5} x^{2}} \] Input:

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

Output:

(b^2*c^4 - 6*a*b*c^2*d^2 - 7*a^2*d^4)*arctan(sqrt(e*x)*d/sqrt(c*d*e))/(sqr 
t(c*d*e)*c^4*d*e^3) - (sqrt(e*x)*b^2*c^4 + 2*sqrt(e*x)*a*b*c^2*d^2 + sqrt( 
e*x)*a^2*d^4)/((d*e*x + c*e)*c^4*d*e^3) - 2/15*(30*a*b*c^2*e^2*x^2 + 45*a^ 
2*d^2*e^2*x^2 - 10*a^2*c*d*e^2*x + 3*a^2*c^2*e^2)/(sqrt(e*x)*c^4*e^5*x^2)
 

Mupad [B] (verification not implemented)

Time = 7.42 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=-\frac {\frac {2\,a^2\,e^3}{5\,c}+\frac {e^3\,x^3\,\left (7\,a^2\,d^4+6\,a\,b\,c^2\,d^2+b^2\,c^4\right )}{c^4\,d}-\frac {14\,a^2\,d\,e^3\,x}{15\,c^2}+\frac {2\,a\,e^3\,x^2\,\left (6\,b\,c^2+7\,a\,d^2\right )}{3\,c^3}}{c\,e^4\,{\left (e\,x\right )}^{5/2}+d\,e^3\,{\left (e\,x\right )}^{7/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e\,x}\,\left (b\,c^2+a\,d^2\right )\,\left (7\,a\,d^2-b\,c^2\right )}{\sqrt {c}\,\sqrt {e}\,\left (7\,a^2\,d^4+6\,a\,b\,c^2\,d^2-b^2\,c^4\right )}\right )\,\left (b\,c^2+a\,d^2\right )\,\left (7\,a\,d^2-b\,c^2\right )}{c^{9/2}\,d^{3/2}\,e^{7/2}} \] Input:

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

Output:

- ((2*a^2*e^3)/(5*c) + (e^3*x^3*(7*a^2*d^4 + b^2*c^4 + 6*a*b*c^2*d^2))/(c^ 
4*d) - (14*a^2*d*e^3*x)/(15*c^2) + (2*a*e^3*x^2*(7*a*d^2 + 6*b*c^2))/(3*c^ 
3))/(c*e^4*(e*x)^(5/2) + d*e^3*(e*x)^(7/2)) - (atan((d^(1/2)*(e*x)^(1/2)*( 
a*d^2 + b*c^2)*(7*a*d^2 - b*c^2))/(c^(1/2)*e^(1/2)*(7*a^2*d^4 - b^2*c^4 + 
6*a*b*c^2*d^2)))*(a*d^2 + b*c^2)*(7*a*d^2 - b*c^2))/(c^(9/2)*d^(3/2)*e^(7/ 
2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} (c+d x)^2} \, dx=\frac {\sqrt {e}\, \left (-105 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c \,d^{4} x^{2}-105 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{5} x^{3}-90 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{3} d^{2} x^{2}-90 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{2} d^{3} x^{3}+15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{5} x^{2}+15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{4} d \,x^{3}-6 a^{2} c^{4} d^{2}+14 a^{2} c^{3} d^{3} x -70 a^{2} c^{2} d^{4} x^{2}-105 a^{2} c \,d^{5} x^{3}-60 a b \,c^{4} d^{2} x^{2}-90 a b \,c^{3} d^{3} x^{3}-15 b^{2} c^{5} d \,x^{3}\right )}{15 \sqrt {x}\, c^{5} d^{2} e^{4} x^{2} \left (d x +c \right )} \] Input:

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

Output:

(sqrt(e)*( - 105*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c) 
))*a**2*c*d**4*x**2 - 105*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d 
)*sqrt(c)))*a**2*d**5*x**3 - 90*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/( 
sqrt(d)*sqrt(c)))*a*b*c**3*d**2*x**2 - 90*sqrt(x)*sqrt(d)*sqrt(c)*atan((sq 
rt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**3*x**3 + 15*sqrt(x)*sqrt(d)*sqrt(c 
)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**5*x**2 + 15*sqrt(x)*sqrt(d)* 
sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**4*d*x**3 - 6*a**2*c**4 
*d**2 + 14*a**2*c**3*d**3*x - 70*a**2*c**2*d**4*x**2 - 105*a**2*c*d**5*x** 
3 - 60*a*b*c**4*d**2*x**2 - 90*a*b*c**3*d**3*x**3 - 15*b**2*c**5*d*x**3))/ 
(15*sqrt(x)*c**5*d**2*e**4*x**2*(c + d*x))