\(\int \frac {(a+b x^2)^2}{(e x)^{3/2} (c+d x)^3} \, dx\) [357]

Optimal result

Integrand size = 24, antiderivative size = 198 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^3} \, dx=-\frac {2 a^2}{c^3 e \sqrt {e x}}+\frac {2 b^2 \sqrt {e x}}{d^3 e^2}-\frac {\left (b c^2+a d^2\right )^2 \sqrt {e x}}{2 c^2 d^3 e^2 (c+d x)^2}+\frac {\left (9 b c^2-7 a d^2\right ) \left (b c^2+a d^2\right ) \sqrt {e x}}{4 c^3 d^3 e^2 (c+d x)}-\frac {\left (15 b^2 c^4-2 a b c^2 d^2+15 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 c^{7/2} d^{7/2} e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^3} \, dx=\frac {x \left (\frac {\sqrt {c} \sqrt {d} \left (-2 a b c^2 d^2 x (c-d x)+b^2 c^3 x \left (15 c^2+25 c d x+8 d^2 x^2\right )-a^2 d^3 \left (8 c^2+25 c d x+15 d^2 x^2\right )\right )}{(c+d x)^2}+\left (-15 b^2 c^4+2 a b c^2 d^2-15 a^2 d^4\right ) \sqrt {x} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )\right )}{4 c^{7/2} d^{7/2} (e x)^{3/2}} \] Input:

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

Output:

(x*((Sqrt[c]*Sqrt[d]*(-2*a*b*c^2*d^2*x*(c - d*x) + b^2*c^3*x*(15*c^2 + 25* 
c*d*x + 8*d^2*x^2) - a^2*d^3*(8*c^2 + 25*c*d*x + 15*d^2*x^2)))/(c + d*x)^2 
 + (-15*b^2*c^4 + 2*a*b*c^2*d^2 - 15*a^2*d^4)*Sqrt[x]*ArcTan[(Sqrt[d]*Sqrt 
[x])/Sqrt[c]]))/(4*c^(7/2)*d^(7/2)*(e*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {518, 1583, 25, 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.

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

\(\Big \downarrow \) 518

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

\(\Big \downarrow \) 1583

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2336

\(\displaystyle \frac {2 \left (\frac {\frac {d e^3 \sqrt {e x} \left (9 b c^2-7 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}-\frac {\int -\frac {8 a^2 c d^4 e^5+8 b^2 c^3 d^2 x^2 e^5-d \left (7 b^2 c^4-2 a b d^2 c^2+7 a^2 d^4\right ) x e^5}{e x (c e+d x e)}d\sqrt {e x}}{2 c e}}{4 c^2 d^4 e^2}-\frac {e^2 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^2 d^3 (c e+d e x)^2}\right )}{e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {8 a^2 c d^4 e^5+8 b^2 c^3 d^2 x^2 e^5-d \left (7 b^2 c^4-2 a b d^2 c^2+7 a^2 d^4\right ) x e^5}{e x (c e+d x e)}d\sqrt {e x}}{2 c e}+\frac {d e^3 \sqrt {e x} \left (9 b c^2-7 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^2 d^4 e^2}-\frac {e^2 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^2 d^3 (c e+d e x)^2}\right )}{e^2}\)

\(\Big \downarrow \) 1584

\(\displaystyle \frac {2 \left (\frac {\frac {\int \left (\frac {8 a^2 e^3 d^4}{x}+8 b^2 c^3 e^3 d-\frac {\left (15 b^2 c^4-2 a b d^2 c^2+15 a^2 d^4\right ) e^4 d}{c e+d x e}\right )d\sqrt {e x}}{2 c e}+\frac {d e^3 \sqrt {e x} \left (9 b c^2-7 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^2 d^4 e^2}-\frac {e^2 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^2 d^3 (c e+d e x)^2}\right )}{e^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (\frac {\frac {-\frac {\sqrt {d} e^{7/2} \left (15 a^2 d^4-2 a b c^2 d^2+15 b^2 c^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {c}}-\frac {8 a^2 d^4 e^4}{\sqrt {e x}}+8 b^2 c^3 d e^3 \sqrt {e x}}{2 c e}+\frac {d e^3 \sqrt {e x} \left (9 b c^2-7 a d^2\right ) \left (a d^2+b c^2\right )}{2 c (c e+d e x)}}{4 c^2 d^4 e^2}-\frac {e^2 \sqrt {e x} \left (a d^2+b c^2\right )^2}{4 c^2 d^3 (c e+d e x)^2}\right )}{e^2}\)

Input:

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

Output:

(2*(-1/4*((b*c^2 + a*d^2)^2*e^2*Sqrt[e*x])/(c^2*d^3*(c*e + d*e*x)^2) + ((d 
*(9*b*c^2 - 7*a*d^2)*(b*c^2 + a*d^2)*e^3*Sqrt[e*x])/(2*c*(c*e + d*e*x)) + 
((-8*a^2*d^4*e^4)/Sqrt[e*x] + 8*b^2*c^3*d*e^3*Sqrt[e*x] - (Sqrt[d]*(15*b^2 
*c^4 - 2*a*b*c^2*d^2 + 15*a^2*d^4)*e^(7/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqr 
t[c]*Sqrt[e])])/Sqrt[c])/(2*c*e))/(4*c^2*d^4*e^2)))/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 = 0.36 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {15 \left (d x +c \right )^{2} \sqrt {e x}\, \left (a^{2} d^{4}-\frac {2}{15} b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8}+\sqrt {d e c}\, \left (-\frac {15 c^{5} b^{2} x}{8}-\frac {25 b^{2} c^{4} d \,x^{2}}{8}+\frac {b \,d^{2} x \left (-4 b \,x^{2}+a \right ) c^{3}}{4}+a \,d^{3} \left (-\frac {b \,x^{2}}{4}+a \right ) c^{2}+\frac {25 a^{2} c \,d^{4} x}{8}+\frac {15 a^{2} x^{2} d^{5}}{8}\right )\right )}{e \sqrt {e x}\, \sqrt {d e c}\, d^{3} c^{3} \left (d x +c \right )^{2}}\)	\(168\)
derivativedivides	\(\frac {\frac {2 b^{2} \sqrt {e x}}{d^{3}}-\frac {2 a^{2} e}{c^{3} \sqrt {e x}}-\frac {2 e \left (\frac {\left (\frac {7}{8} a^{2} d^{5}-\frac {1}{4} d^{3} a \,c^{2} b -\frac {9}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\frac {c e \left (9 a^{2} d^{4}+2 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}-2 b \,c^{2} d^{2} a +15 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{c^{3} d^{3}}}{e^{2}}\)	\(174\)
default	\(\frac {\frac {2 b^{2} \sqrt {e x}}{d^{3}}-\frac {2 a^{2} e}{c^{3} \sqrt {e x}}-\frac {2 e \left (\frac {\left (\frac {7}{8} a^{2} d^{5}-\frac {1}{4} d^{3} a \,c^{2} b -\frac {9}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\frac {c e \left (9 a^{2} d^{4}+2 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) \sqrt {e x}}{8}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}-2 b \,c^{2} d^{2} a +15 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 \sqrt {d e c}}\right )}{c^{3} d^{3}}}{e^{2}}\)	\(174\)
risch	\(-\frac {2 \left (-c^{3} b^{2} x +a^{2} d^{3}\right )}{c^{3} \sqrt {e x}\, d^{3} e}-\frac {\frac {2 \left (\frac {7}{8} a^{2} d^{5}-\frac {1}{4} d^{3} a \,c^{2} b -\frac {9}{8} b^{2} c^{4} d \right ) \left (e x \right )^{\frac {3}{2}}+\frac {c e \left (9 a^{2} d^{4}+2 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) \sqrt {e x}}{4}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}-2 b \,c^{2} d^{2} a +15 b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{4 \sqrt {d e c}}}{d^{3} c^{3} e}\)	\(179\)

Input:

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

Output:

-2/e/(e*x)^(1/2)/(d*e*c)^(1/2)*(15/8*(d*x+c)^2*(e*x)^(1/2)*(a^2*d^4-2/15*b 
*c^2*d^2*a+b^2*c^4)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2))+(d*e*c)^(1/2)*(-15 
/8*c^5*b^2*x-25/8*b^2*c^4*d*x^2+1/4*b*d^2*x*(-4*b*x^2+a)*c^3+a*d^3*(-1/4*b 
*x^2+a)*c^2+25/8*a^2*c*d^4*x+15/8*a^2*x^2*d^5))/d^3/c^3/(d*x+c)^2
 

Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^3} \, dx=\left [-\frac {{\left ({\left (15 \, b^{2} c^{4} d^{2} - 2 \, a b c^{2} d^{4} + 15 \, a^{2} d^{6}\right )} x^{3} + 2 \, {\left (15 \, b^{2} c^{5} d - 2 \, a b c^{3} d^{3} + 15 \, a^{2} c d^{5}\right )} x^{2} + {\left (15 \, b^{2} c^{6} - 2 \, a b c^{4} d^{2} + 15 \, a^{2} c^{2} d^{4}\right )} x\right )} \sqrt {-c d e} \log \left (\frac {d e x - c e + 2 \, \sqrt {-c d e} \sqrt {e x}}{d x + c}\right ) - 2 \, {\left (8 \, b^{2} c^{4} d^{3} x^{3} - 8 \, a^{2} c^{3} d^{4} + {\left (25 \, b^{2} c^{5} d^{2} + 2 \, a b c^{3} d^{4} - 15 \, a^{2} c d^{6}\right )} x^{2} + {\left (15 \, b^{2} c^{6} d - 2 \, a b c^{4} d^{3} - 25 \, a^{2} c^{2} d^{5}\right )} x\right )} \sqrt {e x}}{8 \, {\left (c^{4} d^{6} e^{2} x^{3} + 2 \, c^{5} d^{5} e^{2} x^{2} + c^{6} d^{4} e^{2} x\right )}}, \frac {{\left ({\left (15 \, b^{2} c^{4} d^{2} - 2 \, a b c^{2} d^{4} + 15 \, a^{2} d^{6}\right )} x^{3} + 2 \, {\left (15 \, b^{2} c^{5} d - 2 \, a b c^{3} d^{3} + 15 \, a^{2} c d^{5}\right )} x^{2} + {\left (15 \, b^{2} c^{6} - 2 \, a b c^{4} d^{2} + 15 \, a^{2} c^{2} d^{4}\right )} x\right )} \sqrt {c d e} \arctan \left (\frac {\sqrt {c d e} \sqrt {e x}}{d e x}\right ) + {\left (8 \, b^{2} c^{4} d^{3} x^{3} - 8 \, a^{2} c^{3} d^{4} + {\left (25 \, b^{2} c^{5} d^{2} + 2 \, a b c^{3} d^{4} - 15 \, a^{2} c d^{6}\right )} x^{2} + {\left (15 \, b^{2} c^{6} d - 2 \, a b c^{4} d^{3} - 25 \, a^{2} c^{2} d^{5}\right )} x\right )} \sqrt {e x}}{4 \, {\left (c^{4} d^{6} e^{2} x^{3} + 2 \, c^{5} d^{5} e^{2} x^{2} + c^{6} d^{4} e^{2} x\right )}}\right ] \] Input:

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

Output:

[-1/8*(((15*b^2*c^4*d^2 - 2*a*b*c^2*d^4 + 15*a^2*d^6)*x^3 + 2*(15*b^2*c^5* 
d - 2*a*b*c^3*d^3 + 15*a^2*c*d^5)*x^2 + (15*b^2*c^6 - 2*a*b*c^4*d^2 + 15*a 
^2*c^2*d^4)*x)*sqrt(-c*d*e)*log((d*e*x - c*e + 2*sqrt(-c*d*e)*sqrt(e*x))/( 
d*x + c)) - 2*(8*b^2*c^4*d^3*x^3 - 8*a^2*c^3*d^4 + (25*b^2*c^5*d^2 + 2*a*b 
*c^3*d^4 - 15*a^2*c*d^6)*x^2 + (15*b^2*c^6*d - 2*a*b*c^4*d^3 - 25*a^2*c^2* 
d^5)*x)*sqrt(e*x))/(c^4*d^6*e^2*x^3 + 2*c^5*d^5*e^2*x^2 + c^6*d^4*e^2*x), 
1/4*(((15*b^2*c^4*d^2 - 2*a*b*c^2*d^4 + 15*a^2*d^6)*x^3 + 2*(15*b^2*c^5*d 
- 2*a*b*c^3*d^3 + 15*a^2*c*d^5)*x^2 + (15*b^2*c^6 - 2*a*b*c^4*d^2 + 15*a^2 
*c^2*d^4)*x)*sqrt(c*d*e)*arctan(sqrt(c*d*e)*sqrt(e*x)/(d*e*x)) + (8*b^2*c^ 
4*d^3*x^3 - 8*a^2*c^3*d^4 + (25*b^2*c^5*d^2 + 2*a*b*c^3*d^4 - 15*a^2*c*d^6 
)*x^2 + (15*b^2*c^6*d - 2*a*b*c^4*d^3 - 25*a^2*c^2*d^5)*x)*sqrt(e*x))/(c^4 
*d^6*e^2*x^3 + 2*c^5*d^5*e^2*x^2 + c^6*d^4*e^2*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x^2+a)^2/(e*x)^(3/2)/(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 = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^3} \, dx=-\frac {\frac {8 \, a^{2}}{\sqrt {e x} c^{3}} - \frac {8 \, \sqrt {e x} b^{2}}{d^{3} e} + \frac {{\left (15 \, b^{2} c^{4} - 2 \, a b c^{2} d^{2} + 15 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{\sqrt {c d e} c^{3} d^{3}} - \frac {9 \, \sqrt {e x} b^{2} c^{4} d e x + 2 \, \sqrt {e x} a b c^{2} d^{3} e x - 7 \, \sqrt {e x} a^{2} d^{5} e x + 7 \, \sqrt {e x} b^{2} c^{5} e - 2 \, \sqrt {e x} a b c^{3} d^{2} e - 9 \, \sqrt {e x} a^{2} c d^{4} e}{{\left (d e x + c e\right )}^{2} c^{3} d^{3}}}{4 \, e} \] Input:

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

Output:

-1/4*(8*a^2/(sqrt(e*x)*c^3) - 8*sqrt(e*x)*b^2/(d^3*e) + (15*b^2*c^4 - 2*a* 
b*c^2*d^2 + 15*a^2*d^4)*arctan(sqrt(e*x)*d/sqrt(c*d*e))/(sqrt(c*d*e)*c^3*d 
^3) - (9*sqrt(e*x)*b^2*c^4*d*e*x + 2*sqrt(e*x)*a*b*c^2*d^3*e*x - 7*sqrt(e* 
x)*a^2*d^5*e*x + 7*sqrt(e*x)*b^2*c^5*e - 2*sqrt(e*x)*a*b*c^3*d^2*e - 9*sqr 
t(e*x)*a^2*c*d^4*e)/((d*e*x + c*e)^2*c^3*d^3))/e
 

Mupad [B] (verification not implemented)

Time = 7.61 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^3} \, dx=\frac {2\,b^2\,\sqrt {e\,x}}{d^3\,e^2}-\frac {\frac {2\,a^2\,d^3\,e^2}{c}-\frac {e^2\,x^2\,\left (-15\,a^2\,d^5+2\,a\,b\,c^2\,d^3+9\,b^2\,c^4\,d\right )}{4\,c^3}+\frac {e\,x\,\left (25\,e\,a^2\,d^4+2\,e\,a\,b\,c^2\,d^2-7\,e\,b^2\,c^4\right )}{4\,c^2}}{d^5\,e\,{\left (e\,x\right )}^{5/2}+c^2\,d^3\,e^3\,\sqrt {e\,x}+2\,c\,d^4\,e^2\,{\left (e\,x\right )}^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e\,x}}{\sqrt {c}\,\sqrt {e}}\right )\,\left (15\,a^2\,d^4-2\,a\,b\,c^2\,d^2+15\,b^2\,c^4\right )}{4\,c^{7/2}\,d^{7/2}\,e^{3/2}} \] Input:

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

Output:

(2*b^2*(e*x)^(1/2))/(d^3*e^2) - ((2*a^2*d^3*e^2)/c - (e^2*x^2*(9*b^2*c^4*d 
 - 15*a^2*d^5 + 2*a*b*c^2*d^3))/(4*c^3) + (e*x*(25*a^2*d^4*e - 7*b^2*c^4*e 
 + 2*a*b*c^2*d^2*e))/(4*c^2))/(d^5*e*(e*x)^(5/2) + c^2*d^3*e^3*(e*x)^(1/2) 
 + 2*c*d^4*e^2*(e*x)^(3/2)) - (atan((d^(1/2)*(e*x)^(1/2))/(c^(1/2)*e^(1/2) 
))*(15*a^2*d^4 + 15*b^2*c^4 - 2*a*b*c^2*d^2))/(4*c^(7/2)*d^(7/2)*e^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^3} \, dx=\frac {\sqrt {e}\, \left (-15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c^{2} d^{4}-30 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} c \,d^{5} x -15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{6} x^{2}+2 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{4} d^{2}+4 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{3} d^{3} x +2 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b \,c^{2} d^{4} x^{2}-15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{6}-30 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{5} d x -15 \sqrt {x}\, \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{4} d^{2} x^{2}-8 a^{2} c^{3} d^{4}-25 a^{2} c^{2} d^{5} x -15 a^{2} c \,d^{6} x^{2}-2 a b \,c^{4} d^{3} x +2 a b \,c^{3} d^{4} x^{2}+15 b^{2} c^{6} d x +25 b^{2} c^{5} d^{2} x^{2}+8 b^{2} c^{4} d^{3} x^{3}\right )}{4 \sqrt {x}\, c^{4} d^{4} e^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

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

Output:

(sqrt(e)*( - 15*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)) 
)*a**2*c**2*d**4 - 30*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sq 
rt(c)))*a**2*c*d**5*x - 15*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt( 
d)*sqrt(c)))*a**2*d**6*x**2 + 2*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/( 
sqrt(d)*sqrt(c)))*a*b*c**4*d**2 + 4*sqrt(x)*sqrt(d)*sqrt(c)*atan((sqrt(x)* 
d)/(sqrt(d)*sqrt(c)))*a*b*c**3*d**3*x + 2*sqrt(x)*sqrt(d)*sqrt(c)*atan((sq 
rt(x)*d)/(sqrt(d)*sqrt(c)))*a*b*c**2*d**4*x**2 - 15*sqrt(x)*sqrt(d)*sqrt(c 
)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**6 - 30*sqrt(x)*sqrt(d)*sqrt( 
c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**5*d*x - 15*sqrt(x)*sqrt(d)* 
sqrt(c)*atan((sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**4*d**2*x**2 - 8*a**2*c 
**3*d**4 - 25*a**2*c**2*d**5*x - 15*a**2*c*d**6*x**2 - 2*a*b*c**4*d**3*x + 
 2*a*b*c**3*d**4*x**2 + 15*b**2*c**6*d*x + 25*b**2*c**5*d**2*x**2 + 8*b**2 
*c**4*d**3*x**3))/(4*sqrt(x)*c**4*d**4*e**2*(c**2 + 2*c*d*x + d**2*x**2))