\(\int \frac {(c+d x)^2}{x^4 (a+b x^2)^{3/2}} \, dx\) [1226]

Optimal result

Integrand size = 22, antiderivative size = 146 \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {b \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{a^3 \sqrt {a+b x^2}}-\frac {c^2 \sqrt {a+b x^2}}{3 a^2 x^3}-\frac {c d \sqrt {a+b x^2}}{a^2 x^2}+\frac {\left (5 b c^2-3 a d^2\right ) \sqrt {a+b x^2}}{3 a^3 x}+\frac {3 b c d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a^2 c^2-3 a^2 c d x+4 a b c^2 x^2-3 a^2 d^2 x^2-9 a b c d x^3+8 b^2 c^2 x^4-6 a b d^2 x^4}{3 a^3 x^3 \sqrt {a+b x^2}}-\frac {6 b c d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:

Integrate[(c + d*x)^2/(x^4*(a + b*x^2)^(3/2)),x]
 

Output:

(-(a^2*c^2) - 3*a^2*c*d*x + 4*a*b*c^2*x^2 - 3*a^2*d^2*x^2 - 9*a*b*c*d*x^3 
+ 8*b^2*c^2*x^4 - 6*a*b*d^2*x^4)/(3*a^3*x^3*Sqrt[a + b*x^2]) - (6*b*c*d*Ar 
cTanh[(Sqrt[b]*x)/Sqrt[a] - Sqrt[a + b*x^2]/Sqrt[a]])/a^(5/2)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {532, 25, 2338, 25, 2338, 27, 534, 243, 73, 221}

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.

\(\displaystyle \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 532

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {-9 b c d \int \frac {1}{x \sqrt {b x^2+a}}dx+\frac {\sqrt {a+b x^2} \left (5 b c^2-3 a d^2\right )}{a x}-\frac {3 c d \sqrt {a+b x^2}}{x^2}}{3 a}-\frac {c^2 \sqrt {a+b x^2}}{3 a x^3}}{a}-\frac {b \left (2 a c d-x \left (b c^2-a d^2\right )\right )}{a^3 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {-\frac {9}{2} b c d \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+\frac {\sqrt {a+b x^2} \left (5 b c^2-3 a d^2\right )}{a x}-\frac {3 c d \sqrt {a+b x^2}}{x^2}}{3 a}-\frac {c^2 \sqrt {a+b x^2}}{3 a x^3}}{a}-\frac {b \left (2 a c d-x \left (b c^2-a d^2\right )\right )}{a^3 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {-9 c d \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}+\frac {\sqrt {a+b x^2} \left (5 b c^2-3 a d^2\right )}{a x}-\frac {3 c d \sqrt {a+b x^2}}{x^2}}{3 a}-\frac {c^2 \sqrt {a+b x^2}}{3 a x^3}}{a}-\frac {b \left (2 a c d-x \left (b c^2-a d^2\right )\right )}{a^3 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 221

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

Input:

Int[(c + d*x)^2/(x^4*(a + b*x^2)^(3/2)),x]
 

Output:

-((b*(2*a*c*d - (b*c^2 - a*d^2)*x))/(a^3*Sqrt[a + b*x^2])) + (-1/3*(c^2*Sq 
rt[a + b*x^2])/(a*x^3) + ((-3*c*d*Sqrt[a + b*x^2])/x^2 + ((5*b*c^2 - 3*a*d 
^2)*Sqrt[a + b*x^2])/(a*x) + (9*b*c*d*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sq 
rt[a])/(3*a))/a
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (3 a \,d^{2} x^{2}-5 b \,c^{2} x^{2}+3 a d x c +a \,c^{2}\right )}{3 a^{3} x^{3}}-\frac {b \,d^{2} x}{a^{2} \sqrt {b \,x^{2}+a}}-\frac {2 b d c}{a^{2} \sqrt {b \,x^{2}+a}}+\frac {b^{2} c^{2} x}{a^{3} \sqrt {b \,x^{2}+a}}+\frac {3 b c d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {5}{2}}}\)	\(136\)
default	\(c^{2} \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )+d^{2} \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )+2 c d \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )\)	\(172\)

Input:

int((d*x+c)^2/x^4/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.01 \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {9 \, {\left (b^{2} c d x^{5} + a b c d x^{3}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (9 \, a b c d x^{3} + 3 \, a^{2} c d x - 2 \, {\left (4 \, b^{2} c^{2} - 3 \, a b d^{2}\right )} x^{4} + a^{2} c^{2} - {\left (4 \, a b c^{2} - 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, -\frac {9 \, {\left (b^{2} c d x^{5} + a b c d x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (9 \, a b c d x^{3} + 3 \, a^{2} c d x - 2 \, {\left (4 \, b^{2} c^{2} - 3 \, a b d^{2}\right )} x^{4} + a^{2} c^{2} - {\left (4 \, a b c^{2} - 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \] Input:

integrate((d*x+c)^2/x^4/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/6*(9*(b^2*c*d*x^5 + a*b*c*d*x^3)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a 
)*sqrt(a) + 2*a)/x^2) - 2*(9*a*b*c*d*x^3 + 3*a^2*c*d*x - 2*(4*b^2*c^2 - 3* 
a*b*d^2)*x^4 + a^2*c^2 - (4*a*b*c^2 - 3*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a^ 
3*b*x^5 + a^4*x^3), -1/3*(9*(b^2*c*d*x^5 + a*b*c*d*x^3)*sqrt(-a)*arctan(sq 
rt(b*x^2 + a)*sqrt(-a)/a) + (9*a*b*c*d*x^3 + 3*a^2*c*d*x - 2*(4*b^2*c^2 - 
3*a*b*d^2)*x^4 + a^2*c^2 - (4*a*b*c^2 - 3*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/( 
a^3*b*x^5 + a^4*x^3)]
 

Sympy [F]

\[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{x^{4} \left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((d*x+c)**2/x**4/(b*x**2+a)**(3/2),x)
 

Output:

Integral((c + d*x)**2/(x**4*(a + b*x**2)**(3/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=\frac {8 \, b^{2} c^{2} x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {2 \, b d^{2} x}{\sqrt {b x^{2} + a} a^{2}} + \frac {3 \, b c d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} - \frac {3 \, b c d}{\sqrt {b x^{2} + a} a^{2}} + \frac {4 \, b c^{2}}{3 \, \sqrt {b x^{2} + a} a^{2} x} - \frac {d^{2}}{\sqrt {b x^{2} + a} a x} - \frac {c d}{\sqrt {b x^{2} + a} a x^{2}} - \frac {c^{2}}{3 \, \sqrt {b x^{2} + a} a x^{3}} \] Input:

integrate((d*x+c)^2/x^4/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

8/3*b^2*c^2*x/(sqrt(b*x^2 + a)*a^3) - 2*b*d^2*x/(sqrt(b*x^2 + a)*a^2) + 3* 
b*c*d*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 3*b*c*d/(sqrt(b*x^2 + a)*a^2 
) + 4/3*b*c^2/(sqrt(b*x^2 + a)*a^2*x) - d^2/(sqrt(b*x^2 + a)*a*x) - c*d/(s 
qrt(b*x^2 + a)*a*x^2) - 1/3*c^2/(sqrt(b*x^2 + a)*a*x^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (128) = 256\).

Time = 0.13 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.05 \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {6 \, b c d \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {\frac {2 \, b c d}{a^{2}} - \frac {{\left (a^{2} b^{2} c^{2} - a^{3} b d^{2}\right )} x}{a^{5}}}{\sqrt {b x^{2} + a}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} b c d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} c^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a \sqrt {b} d^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c d - 5 \, a^{2} b^{\frac {3}{2}} c^{2} + 3 \, a^{3} \sqrt {b} d^{2}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \] Input:

integrate((d*x+c)^2/x^4/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

-6*b*c*d*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 
(2*b*c*d/a^2 - (a^2*b^2*c^2 - a^3*b*d^2)*x/a^5)/sqrt(b*x^2 + a) + 2/3*(3*( 
sqrt(b)*x - sqrt(b*x^2 + a))^5*b*c*d - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b 
^(3/2)*c^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*sqrt(b)*d^2 + 12*(sqrt(b) 
*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c^2 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2* 
a^2*sqrt(b)*d^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b*c*d - 5*a^2*b^(3/2 
)*c^2 + 3*a^3*sqrt(b)*d^2)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{x^4\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int((c + d*x)^2/(x^4*(a + b*x^2)^(3/2)),x)
 

Output:

int((c + d*x)^2/(x^4*(a + b*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.23 \[ \int \frac {(c+d x)^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a^{2} c^{2}-3 \sqrt {b \,x^{2}+a}\, a^{2} c d x -3 \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x^{2}+4 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{2}-9 \sqrt {b \,x^{2}+a}\, a b c d \,x^{3}-6 \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{4}+8 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x^{4}-9 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{3}-9 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{5}+9 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{3}+9 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{5}+6 \sqrt {b}\, a^{2} d^{2} x^{3}-8 \sqrt {b}\, a b \,c^{2} x^{3}+6 \sqrt {b}\, a b \,d^{2} x^{5}-8 \sqrt {b}\, b^{2} c^{2} x^{5}}{3 a^{3} x^{3} \left (b \,x^{2}+a \right )} \] Input:

int((d*x+c)^2/x^4/(b*x^2+a)^(3/2),x)
 

Output:

( - sqrt(a + b*x**2)*a**2*c**2 - 3*sqrt(a + b*x**2)*a**2*c*d*x - 3*sqrt(a 
+ b*x**2)*a**2*d**2*x**2 + 4*sqrt(a + b*x**2)*a*b*c**2*x**2 - 9*sqrt(a + b 
*x**2)*a*b*c*d*x**3 - 6*sqrt(a + b*x**2)*a*b*d**2*x**4 + 8*sqrt(a + b*x**2 
)*b**2*c**2*x**4 - 9*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/ 
sqrt(a))*a*b*c*d*x**3 - 9*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b 
)*x)/sqrt(a))*b**2*c*d*x**5 + 9*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + 
sqrt(b)*x)/sqrt(a))*a*b*c*d*x**3 + 9*sqrt(a)*log((sqrt(a + b*x**2) + sqrt( 
a) + sqrt(b)*x)/sqrt(a))*b**2*c*d*x**5 + 6*sqrt(b)*a**2*d**2*x**3 - 8*sqrt 
(b)*a*b*c**2*x**3 + 6*sqrt(b)*a*b*d**2*x**5 - 8*sqrt(b)*b**2*c**2*x**5)/(3 
*a**3*x**3*(a + b*x**2))