\(\int \frac {(a+b x^2)^2}{(e x)^{5/2} (c+d x)^3} \, dx\) [358]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=-\frac {2 a^2}{3 c^3 e (e x)^{3/2}}+\frac {6 a^2 d}{c^4 e^2 \sqrt {e x}}+\frac {\left (b c^2+a d^2\right )^2 \sqrt {e x}}{2 c^3 d^2 e^3 (c+d x)^2}-\frac {\left (5 b c^2-11 a d^2\right ) \left (b c^2+a d^2\right ) \sqrt {e x}}{4 c^4 d^2 e^3 (c+d x)}+\frac {\left (3 b^2 c^4+6 a b c^2 d^2+35 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 c^{9/2} d^{5/2} e^{5/2}} \] Output:

-2/3*a^2/c^3/e/(e*x)^(3/2)+6*a^2*d/c^4/e^2/(e*x)^(1/2)+1/2*(a*d^2+b*c^2)^2 
*(e*x)^(1/2)/c^3/d^2/e^3/(d*x+c)^2-1/4*(-11*a*d^2+5*b*c^2)*(a*d^2+b*c^2)*( 
e*x)^(1/2)/c^4/d^2/e^3/(d*x+c)+1/4*(35*a^2*d^4+6*a*b*c^2*d^2+3*b^2*c^4)*ar 
ctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/c^(9/2)/d^(5/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\frac {x \left (\frac {\sqrt {c} \sqrt {d} \left (6 a b c^2 d^2 x^2 (5 c+3 d x)-3 b^2 c^4 x^2 (3 c+5 d x)+a^2 d^2 \left (-8 c^3+56 c^2 d x+175 c d^2 x^2+105 d^3 x^3\right )\right )}{(c+d x)^2}+3 \left (3 b^2 c^4+6 a b c^2 d^2+35 a^2 d^4\right ) x^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )\right )}{12 c^{9/2} d^{5/2} (e x)^{5/2}} \] Input:

Integrate[(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x)^3),x]
 

Output:

(x*((Sqrt[c]*Sqrt[d]*(6*a*b*c^2*d^2*x^2*(5*c + 3*d*x) - 3*b^2*c^4*x^2*(3*c 
 + 5*d*x) + a^2*d^2*(-8*c^3 + 56*c^2*d*x + 175*c*d^2*x^2 + 105*d^3*x^3)))/ 
(c + d*x)^2 + 3*(3*b^2*c^4 + 6*a*b*c^2*d^2 + 35*a^2*d^4)*x^(3/2)*ArcTan[(S 
qrt[d]*Sqrt[x])/Sqrt[c]]))/(12*c^(9/2)*d^(5/2)*(e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {518, 1583, 2336, 25, 1584, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx\)

\(\Big \downarrow \) 518

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

\(\Big \downarrow \) 1583

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

\(\Big \downarrow \) 2336

\(\displaystyle \frac {2 \left (\frac {-\frac {\int -\frac {8 a^2 c^2 d^4 e^6+d^2 \left (3 b^2 c^4+6 a b d^2 c^2+11 a^2 d^4\right ) x^2 e^6-16 a^2 c d^5 x e^6}{e^2 x^2 (c e+d x e)}d\sqrt {e x}}{2 c e}-\frac {d^2 e^3 \sqrt {e x} \left (5 b c^2-11 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^3 d^4 e^3}+\frac {e \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^3 d^2 (c e+d e x)^2}\right )}{e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {8 a^2 c^2 d^4 e^6+d^2 \left (3 b^2 c^4+6 a b d^2 c^2+11 a^2 d^4\right ) x^2 e^6-16 a^2 c d^5 x e^6}{e^2 x^2 (c e+d x e)}d\sqrt {e x}}{2 c e}-\frac {d^2 e^3 \sqrt {e x} \left (5 b c^2-11 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^3 d^4 e^3}+\frac {e \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^3 d^2 (c e+d e x)^2}\right )}{e^2}\)

\(\Big \downarrow \) 1584

\(\displaystyle \frac {2 \left (\frac {\frac {\int \left (-\frac {24 a^2 e^3 d^5}{x}+\frac {8 a^2 c e^3 d^4}{x^2}+\frac {\left (3 b^2 c^4+6 a b d^2 c^2+35 a^2 d^4\right ) e^4 d^2}{c e+d x e}\right )d\sqrt {e x}}{2 c e}-\frac {d^2 e^3 \sqrt {e x} \left (5 b c^2-11 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^3 d^4 e^3}+\frac {e \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^3 d^2 (c e+d e x)^2}\right )}{e^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {\frac {\frac {d^{3/2} e^{7/2} \left (35 a^2 d^4+6 a b c^2 d^2+3 b^2 c^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c}}-\frac {8 a^2 c d^4 e^5}{3 (e x)^{3/2}}+\frac {24 a^2 d^5 e^4}{\sqrt {e x}}}{2 c e}-\frac {d^2 e^3 \sqrt {e x} \left (5 b c^2-11 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^3 d^4 e^3}+\frac {e \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^3 d^2 (c e+d e x)^2}\right )}{e^2}\)

Input:

Int[(a + b*x^2)^2/((e*x)^(5/2)*(c + d*x)^3),x]
 

Output:

(2*(((b*c^2 + a*d^2)^2*e*Sqrt[e*x])/(4*c^3*d^2*(c*e + d*e*x)^2) + (-1/2*(d 
^2*(5*b*c^2 - 11*a*d^2)*(b*c^2 + a*d^2)*e^3*Sqrt[e*x])/(c*(c*e + d*e*x)) + 
 ((-8*a^2*c*d^4*e^5)/(3*(e*x)^(3/2)) + (24*a^2*d^5*e^4)/Sqrt[e*x] + (d^(3/ 
2)*(3*b^2*c^4 + 6*a*b*c^2*d^2 + 35*a^2*d^4)*e^(7/2)*ArcTan[(Sqrt[d]*Sqrt[e 
*x])/(Sqrt[c]*Sqrt[e])])/Sqrt[c])/(2*c*e))/(4*c^3*d^4*e^3)))/e^2
 

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 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

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

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88

method	result	size
risch	\(-\frac {2 a^{2} \left (-9 d x +c \right )}{3 c^{4} \sqrt {e x}\, x \,e^{2}}+\frac {\frac {\frac {\left (11 a^{2} d^{4}+6 b \,c^{2} d^{2} a -5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{4 d}+\frac {c e \left (13 a^{2} d^{4}+10 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right ) \sqrt {e x}}{4 d^{2}}}{\left (d e x +c e \right )^{2}}+\frac {\left (35 a^{2} d^{4}+6 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{4 d^{2} \sqrt {d e c}}}{c^{4} e^{2}}\)	\(176\)
pseudoelliptic	\(-\frac {2 \left (-\frac {105 \left (d x +c \right )^{2} x \sqrt {e x}\, \left (a^{2} d^{4}+\frac {6}{35} b \,c^{2} d^{2} a +\frac {3}{35} b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8}+\sqrt {d e c}\, \left (-\frac {105 a^{2} d^{5} x^{3}}{8}-\frac {175 a^{2} c \,d^{4} x^{2}}{8}-7 x \left (\frac {9 b \,x^{2}}{28}+a \right ) a \,c^{2} d^{3}+a \,c^{3} \left (-\frac {15 b \,x^{2}}{4}+a \right ) d^{2}+\frac {15 b^{2} c^{4} d \,x^{3}}{8}+\frac {9 b^{2} c^{5} x^{2}}{8}\right )\right )}{3 \sqrt {e x}\, \sqrt {d e c}\, e^{2} c^{4} x \left (d x +c \right )^{2} d^{2}}\)	\(177\)
derivativedivides	\(\frac {-\frac {2 a^{2} e}{3 c^{3} \left (e x \right )^{\frac {3}{2}}}+\frac {6 a^{2} d}{c^{4} \sqrt {e x}}+\frac {2 \left (\frac {\frac {\left (11 a^{2} d^{4}+6 b \,c^{2} d^{2} a -5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{8 d}+\frac {c e \left (13 a^{2} d^{4}+10 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right ) \sqrt {e x}}{8 d^{2}}}{\left (d e x +c e \right )^{2}}+\frac {\left (35 a^{2} d^{4}+6 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 d^{2} \sqrt {d e c}}\right )}{c^{4}}}{e^{2}}\)	\(180\)
default	\(\frac {-\frac {2 a^{2} e}{3 c^{3} \left (e x \right )^{\frac {3}{2}}}+\frac {6 a^{2} d}{c^{4} \sqrt {e x}}+\frac {2 \left (\frac {\frac {\left (11 a^{2} d^{4}+6 b \,c^{2} d^{2} a -5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{8 d}+\frac {c e \left (13 a^{2} d^{4}+10 b \,c^{2} d^{2} a -3 b^{2} c^{4}\right ) \sqrt {e x}}{8 d^{2}}}{\left (d e x +c e \right )^{2}}+\frac {\left (35 a^{2} d^{4}+6 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 d^{2} \sqrt {d e c}}\right )}{c^{4}}}{e^{2}}\)	\(180\)

Input:

int((b*x^2+a)^2/(e*x)^(5/2)/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-2/3*a^2*(-9*d*x+c)/c^4/(e*x)^(1/2)/x/e^2+1/c^4*(2*(1/8*(11*a^2*d^4+6*a*b* 
c^2*d^2-5*b^2*c^4)/d*(e*x)^(3/2)+1/8*c*e*(13*a^2*d^4+10*a*b*c^2*d^2-3*b^2* 
c^4)/d^2*(e*x)^(1/2))/(d*e*x+c*e)^2+1/4*(35*a^2*d^4+6*a*b*c^2*d^2+3*b^2*c^ 
4)/d^2/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)))/e^2
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\left [-\frac {3 \, {\left ({\left (3 \, b^{2} c^{4} d^{2} + 6 \, a b c^{2} d^{4} + 35 \, a^{2} d^{6}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{5} d + 6 \, a b c^{3} d^{3} + 35 \, a^{2} c d^{5}\right )} x^{3} + {\left (3 \, b^{2} c^{6} + 6 \, a b c^{4} d^{2} + 35 \, a^{2} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {-c d e} \log \left (\frac {d e x - c e - 2 \, \sqrt {-c d e} \sqrt {e x}}{d x + c}\right ) - 2 \, {\left (56 \, a^{2} c^{3} d^{4} x - 8 \, a^{2} c^{4} d^{3} - 3 \, {\left (5 \, b^{2} c^{5} d^{2} - 6 \, a b c^{3} d^{4} - 35 \, a^{2} c d^{6}\right )} x^{3} - {\left (9 \, b^{2} c^{6} d - 30 \, a b c^{4} d^{3} - 175 \, a^{2} c^{2} d^{5}\right )} x^{2}\right )} \sqrt {e x}}{24 \, {\left (c^{5} d^{5} e^{3} x^{4} + 2 \, c^{6} d^{4} e^{3} x^{3} + c^{7} d^{3} e^{3} x^{2}\right )}}, -\frac {3 \, {\left ({\left (3 \, b^{2} c^{4} d^{2} + 6 \, a b c^{2} d^{4} + 35 \, a^{2} d^{6}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{5} d + 6 \, a b c^{3} d^{3} + 35 \, a^{2} c d^{5}\right )} x^{3} + {\left (3 \, b^{2} c^{6} + 6 \, a b c^{4} d^{2} + 35 \, a^{2} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {c d e} \arctan \left (\frac {\sqrt {c d e} \sqrt {e x}}{d e x}\right ) - {\left (56 \, a^{2} c^{3} d^{4} x - 8 \, a^{2} c^{4} d^{3} - 3 \, {\left (5 \, b^{2} c^{5} d^{2} - 6 \, a b c^{3} d^{4} - 35 \, a^{2} c d^{6}\right )} x^{3} - {\left (9 \, b^{2} c^{6} d - 30 \, a b c^{4} d^{3} - 175 \, a^{2} c^{2} d^{5}\right )} x^{2}\right )} \sqrt {e x}}{12 \, {\left (c^{5} d^{5} e^{3} x^{4} + 2 \, c^{6} d^{4} e^{3} x^{3} + c^{7} d^{3} e^{3} x^{2}\right )}}\right ] \] Input:

integrate((b*x^2+a)^2/(e*x)^(5/2)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

[-1/24*(3*((3*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + 35*a^2*d^6)*x^4 + 2*(3*b^2*c^5 
*d + 6*a*b*c^3*d^3 + 35*a^2*c*d^5)*x^3 + (3*b^2*c^6 + 6*a*b*c^4*d^2 + 35*a 
^2*c^2*d^4)*x^2)*sqrt(-c*d*e)*log((d*e*x - c*e - 2*sqrt(-c*d*e)*sqrt(e*x)) 
/(d*x + c)) - 2*(56*a^2*c^3*d^4*x - 8*a^2*c^4*d^3 - 3*(5*b^2*c^5*d^2 - 6*a 
*b*c^3*d^4 - 35*a^2*c*d^6)*x^3 - (9*b^2*c^6*d - 30*a*b*c^4*d^3 - 175*a^2*c 
^2*d^5)*x^2)*sqrt(e*x))/(c^5*d^5*e^3*x^4 + 2*c^6*d^4*e^3*x^3 + c^7*d^3*e^3 
*x^2), -1/12*(3*((3*b^2*c^4*d^2 + 6*a*b*c^2*d^4 + 35*a^2*d^6)*x^4 + 2*(3*b 
^2*c^5*d + 6*a*b*c^3*d^3 + 35*a^2*c*d^5)*x^3 + (3*b^2*c^6 + 6*a*b*c^4*d^2 
+ 35*a^2*c^2*d^4)*x^2)*sqrt(c*d*e)*arctan(sqrt(c*d*e)*sqrt(e*x)/(d*e*x)) - 
 (56*a^2*c^3*d^4*x - 8*a^2*c^4*d^3 - 3*(5*b^2*c^5*d^2 - 6*a*b*c^3*d^4 - 35 
*a^2*c*d^6)*x^3 - (9*b^2*c^6*d - 30*a*b*c^4*d^3 - 175*a^2*c^2*d^5)*x^2)*sq 
rt(e*x))/(c^5*d^5*e^3*x^4 + 2*c^6*d^4*e^3*x^3 + c^7*d^3*e^3*x^2)]
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {5}{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate((b*x**2+a)**2/(e*x)**(5/2)/(d*x+c)**3,x)
 

Output:

Integral((a + b*x**2)**2/((e*x)**(5/2)*(c + d*x)**3), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)^2/(e*x)^(5/2)/(d*x+c)^3,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 = 0.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\frac {{\left (3 \, b^{2} c^{4} + 6 \, a b c^{2} d^{2} + 35 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{4 \, \sqrt {c d e} c^{4} d^{2} e^{2}} + \frac {2 \, {\left (9 \, a^{2} d e x - a^{2} c e\right )}}{3 \, \sqrt {e x} c^{4} e^{3} x} - \frac {5 \, \sqrt {e x} b^{2} c^{4} d e x - 6 \, \sqrt {e x} a b c^{2} d^{3} e x - 11 \, \sqrt {e x} a^{2} d^{5} e x + 3 \, \sqrt {e x} b^{2} c^{5} e - 10 \, \sqrt {e x} a b c^{3} d^{2} e - 13 \, \sqrt {e x} a^{2} c d^{4} e}{4 \, {\left (d e x + c e\right )}^{2} c^{4} d^{2} e^{2}} \] Input:

integrate((b*x^2+a)^2/(e*x)^(5/2)/(d*x+c)^3,x, algorithm="giac")
 

Output:

1/4*(3*b^2*c^4 + 6*a*b*c^2*d^2 + 35*a^2*d^4)*arctan(sqrt(e*x)*d/sqrt(c*d*e 
))/(sqrt(c*d*e)*c^4*d^2*e^2) + 2/3*(9*a^2*d*e*x - a^2*c*e)/(sqrt(e*x)*c^4* 
e^3*x) - 1/4*(5*sqrt(e*x)*b^2*c^4*d*e*x - 6*sqrt(e*x)*a*b*c^2*d^3*e*x - 11 
*sqrt(e*x)*a^2*d^5*e*x + 3*sqrt(e*x)*b^2*c^5*e - 10*sqrt(e*x)*a*b*c^3*d^2* 
e - 13*sqrt(e*x)*a^2*c*d^4*e)/((d*e*x + c*e)^2*c^4*d^2*e^2)
 

Mupad [B] (verification not implemented)

Time = 7.57 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\frac {\frac {e^3\,x^3\,\left (35\,a^2\,d^4+6\,a\,b\,c^2\,d^2-5\,b^2\,c^4\right )}{4\,c^4\,d}-\frac {2\,a^2\,e^3}{3\,c}+\frac {e^3\,x^2\,\left (175\,a^2\,d^4+30\,a\,b\,c^2\,d^2-9\,b^2\,c^4\right )}{12\,c^3\,d^2}+\frac {14\,a^2\,d\,e^3\,x}{3\,c^2}}{c^2\,e^4\,{\left (e\,x\right )}^{3/2}+d^2\,e^2\,{\left (e\,x\right )}^{7/2}+2\,c\,d\,e^3\,{\left (e\,x\right )}^{5/2}}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e\,x}}{\sqrt {c}\,\sqrt {e}}\right )\,\left (35\,a^2\,d^4+6\,a\,b\,c^2\,d^2+3\,b^2\,c^4\right )}{4\,c^{9/2}\,d^{5/2}\,e^{5/2}} \] Input:

int((a + b*x^2)^2/((e*x)^(5/2)*(c + d*x)^3),x)
 

Output:

((e^3*x^3*(35*a^2*d^4 - 5*b^2*c^4 + 6*a*b*c^2*d^2))/(4*c^4*d) - (2*a^2*e^3 
)/(3*c) + (e^3*x^2*(175*a^2*d^4 - 9*b^2*c^4 + 30*a*b*c^2*d^2))/(12*c^3*d^2 
) + (14*a^2*d*e^3*x)/(3*c^2))/(c^2*e^4*(e*x)^(3/2) + d^2*e^2*(e*x)^(7/2) + 
 2*c*d*e^3*(e*x)^(5/2)) + (atan((d^(1/2)*(e*x)^(1/2))/(c^(1/2)*e^(1/2)))*( 
35*a^2*d^4 + 3*b^2*c^4 + 6*a*b*c^2*d^2))/(4*c^(9/2)*d^(5/2)*e^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.08 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} (c+d x)^3} \, dx=\frac {\sqrt {e}\, \left (105 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c^{2} d^{4} x +210 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c \,d^{5} x^{2}+105 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{6} x^{3}+18 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{4} d^{2} x +36 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{3} d^{3} x^{2}+18 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{2} d^{4} x^{3}+9 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{6} x +18 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{5} d \,x^{2}+9 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{4} d^{2} x^{3}-8 a^{2} c^{4} d^{3}+56 a^{2} c^{3} d^{4} x +175 a^{2} c^{2} d^{5} x^{2}+105 a^{2} c \,d^{6} x^{3}+30 a b \,c^{4} d^{3} x^{2}+18 a b \,c^{3} d^{4} x^{3}-9 b^{2} c^{6} d \,x^{2}-15 b^{2} c^{5} d^{2} x^{3}\right )}{12 \sqrt {x}\, c^{5} d^{3} e^{3} x \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int((b*x^2+a)^2/(e*x)^(5/2)/(d*x+c)^3,x)
 

Output:

(sqrt(e)*(105*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))* 
a**2*c**2*d**4*x + 210*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*s 
qrt(c)))*a**2*c*d**5*x**2 + 105*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/( 
sqrt(d)*sqrt(c)))*a**2*d**6*x**3 + 18*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x 
)*d)/(sqrt(d)*sqrt(c)))*a*b*c**4*d**2*x + 36*sqrt(x)*sqrt(d)*sqrt(c)*atan( 
(sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**3*d**3*x**2 + 18*sqrt(x)*sqrt(d)*sqr 
t(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**4*x**3 + 9*sqrt(x)*sq 
rt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**6*x + 18*sqrt(x) 
*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**5*d*x**2 + 9* 
sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**4*d**2 
*x**3 - 8*a**2*c**4*d**3 + 56*a**2*c**3*d**4*x + 175*a**2*c**2*d**5*x**2 + 
 105*a**2*c*d**6*x**3 + 30*a*b*c**4*d**3*x**2 + 18*a*b*c**3*d**4*x**3 - 9* 
b**2*c**6*d*x**2 - 15*b**2*c**5*d**2*x**3))/(12*sqrt(x)*c**5*d**3*e**3*x*( 
c**2 + 2*c*d*x + d**2*x**2))