\(\int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx\) [1179]

Optimal result

Integrand size = 22, antiderivative size = 173 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}-\frac {c d \sqrt {a+b x^2}}{2 a x^4}+\frac {\left (4 b c^2-5 a d^2\right ) \sqrt {a+b x^2}}{15 a^2 x^3}+\frac {3 b c d \sqrt {a+b x^2}}{4 a^2 x^2}-\frac {2 b \left (4 b c^2-5 a d^2\right ) \sqrt {a+b x^2}}{15 a^3 x}-\frac {3 b^2 c d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Output:

-1/5*c^2*(b*x^2+a)^(1/2)/a/x^5-1/2*c*d*(b*x^2+a)^(1/2)/a/x^4+1/15*(-5*a*d^ 
2+4*b*c^2)*(b*x^2+a)^(1/2)/a^2/x^3+3/4*b*c*d*(b*x^2+a)^(1/2)/a^2/x^2-2/15* 
b*(-5*a*d^2+4*b*c^2)*(b*x^2+a)^(1/2)/a^3/x-3/4*b^2*c*d*arctanh((b*x^2+a)^( 
1/2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.72 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=\frac {\frac {\sqrt {a+b x^2} \left (-32 b^2 c^2 x^4-2 a^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )+a b x^2 \left (16 c^2+45 c d x+40 d^2 x^2\right )\right )}{x^5}+90 \sqrt {a} b^2 c d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{60 a^3} \] Input:

Integrate[(c + d*x)^2/(x^6*Sqrt[a + b*x^2]),x]
 

Output:

((Sqrt[a + b*x^2]*(-32*b^2*c^2*x^4 - 2*a^2*(6*c^2 + 15*c*d*x + 10*d^2*x^2) 
 + a*b*x^2*(16*c^2 + 45*c*d*x + 40*d^2*x^2)))/x^5 + 90*Sqrt[a]*b^2*c*d*Arc 
Tanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(60*a^3)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {540, 25, 539, 27, 539, 25, 27, 539, 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.

\(\displaystyle \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx\)

\(\Big \downarrow \) 540

\(\displaystyle -\frac {\int -\frac {10 a c d-\left (4 b c^2-5 a d^2\right ) x}{x^5 \sqrt {b x^2+a}}dx}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {10 a c d-\left (4 b c^2-5 a d^2\right ) x}{x^5 \sqrt {b x^2+a}}dx}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {-\frac {\int \frac {2 a \left (2 \left (4 b c^2-5 a d^2\right )+15 b c d x\right )}{x^4 \sqrt {b x^2+a}}dx}{4 a}-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{2} \int \frac {2 \left (4 b c^2-5 a d^2\right )+15 b c d x}{x^4 \sqrt {b x^2+a}}dx-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{2} \left (\frac {\int -\frac {b \left (45 a c d-4 \left (4 b c^2-5 a d^2\right ) x\right )}{x^3 \sqrt {b x^2+a}}dx}{3 a}+\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {\int \frac {b \left (45 a c d-4 \left (4 b c^2-5 a d^2\right ) x\right )}{x^3 \sqrt {b x^2+a}}dx}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \int \frac {45 a c d-4 \left (4 b c^2-5 a d^2\right ) x}{x^3 \sqrt {b x^2+a}}dx}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \left (-\frac {\int \frac {a \left (8 \left (4 b c^2-5 a d^2\right )+45 b c d x\right )}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {45 c d \sqrt {a+b x^2}}{2 x^2}\right )}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \left (-\frac {1}{2} \int \frac {8 \left (4 b c^2-5 a d^2\right )+45 b c d x}{x^2 \sqrt {b x^2+a}}dx-\frac {45 c d \sqrt {a+b x^2}}{2 x^2}\right )}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \left (\frac {1}{2} \left (\frac {8 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{a x}-45 b c d \int \frac {1}{x \sqrt {b x^2+a}}dx\right )-\frac {45 c d \sqrt {a+b x^2}}{2 x^2}\right )}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \left (\frac {1}{2} \left (\frac {8 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{a x}-\frac {45}{2} b c d \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2\right )-\frac {45 c d \sqrt {a+b x^2}}{2 x^2}\right )}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \left (\frac {1}{2} \left (\frac {8 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{a x}-45 c d \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}\right )-\frac {45 c d \sqrt {a+b x^2}}{2 x^2}\right )}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{3 a x^3}-\frac {b \left (\frac {1}{2} \left (\frac {45 b c d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {8 \sqrt {a+b x^2} \left (4 b c^2-5 a d^2\right )}{a x}\right )-\frac {45 c d \sqrt {a+b x^2}}{2 x^2}\right )}{3 a}\right )-\frac {5 c d \sqrt {a+b x^2}}{2 x^4}}{5 a}-\frac {c^2 \sqrt {a+b x^2}}{5 a x^5}\)

Input:

Int[(c + d*x)^2/(x^6*Sqrt[a + b*x^2]),x]
 

Output:

-1/5*(c^2*Sqrt[a + b*x^2])/(a*x^5) + ((-5*c*d*Sqrt[a + b*x^2])/(2*x^4) + ( 
(2*(4*b*c^2 - 5*a*d^2)*Sqrt[a + b*x^2])/(3*a*x^3) - (b*((-45*c*d*Sqrt[a + 
b*x^2])/(2*x^2) + ((8*(4*b*c^2 - 5*a*d^2)*Sqrt[a + b*x^2])/(a*x) + (45*b*c 
*d*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/2))/(3*a))/2)/(5*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 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 539

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

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-40 a b \,d^{2} x^{4}+32 b^{2} c^{2} x^{4}-45 a b c d \,x^{3}+20 a^{2} d^{2} x^{2}-16 a b \,c^{2} x^{2}+30 a^{2} c d x +12 a^{2} c^{2}\right )}{60 a^{3} x^{5}}-\frac {3 c d \,b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{4 a^{\frac {5}{2}}}\)	\(120\)
default	\(c^{2} \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )+d^{2} \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )+2 c d \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\)	\(181\)

Input:

int((d*x+c)^2/x^6/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/60*(b*x^2+a)^(1/2)*(-40*a*b*d^2*x^4+32*b^2*c^2*x^4-45*a*b*c*d*x^3+20*a^ 
2*d^2*x^2-16*a*b*c^2*x^2+30*a^2*c*d*x+12*a^2*c^2)/a^3/x^5-3/4*c*d/a^(5/2)* 
b^2*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 = 252, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=\left [\frac {45 \, \sqrt {a} b^{2} c d x^{5} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (45 \, a b c d x^{3} - 30 \, a^{2} c d x - 8 \, {\left (4 \, b^{2} c^{2} - 5 \, a b d^{2}\right )} x^{4} - 12 \, a^{2} c^{2} + 4 \, {\left (4 \, a b c^{2} - 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{120 \, a^{3} x^{5}}, \frac {45 \, \sqrt {-a} b^{2} c d x^{5} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (45 \, a b c d x^{3} - 30 \, a^{2} c d x - 8 \, {\left (4 \, b^{2} c^{2} - 5 \, a b d^{2}\right )} x^{4} - 12 \, a^{2} c^{2} + 4 \, {\left (4 \, a b c^{2} - 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{60 \, a^{3} x^{5}}\right ] \] Input:

integrate((d*x+c)^2/x^6/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/120*(45*sqrt(a)*b^2*c*d*x^5*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2 
*a)/x^2) + 2*(45*a*b*c*d*x^3 - 30*a^2*c*d*x - 8*(4*b^2*c^2 - 5*a*b*d^2)*x^ 
4 - 12*a^2*c^2 + 4*(4*a*b*c^2 - 5*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3*x^5) 
, 1/60*(45*sqrt(-a)*b^2*c*d*x^5*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (45*a 
*b*c*d*x^3 - 30*a^2*c*d*x - 8*(4*b^2*c^2 - 5*a*b*d^2)*x^4 - 12*a^2*c^2 + 4 
*(4*a*b*c^2 - 5*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3*x^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (163) = 326\).

Time = 3.78 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=- \frac {3 a^{4} b^{\frac {9}{2}} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {2 a^{3} b^{\frac {11}{2}} c^{2} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {3 a^{2} b^{\frac {13}{2}} c^{2} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {12 a b^{\frac {15}{2}} c^{2} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {8 b^{\frac {17}{2}} c^{2} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {c d}{2 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} c d}{4 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {\sqrt {b} d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {3 b^{\frac {3}{2}} c d}{4 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {2 b^{\frac {3}{2}} d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} - \frac {3 b^{2} c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{4 a^{\frac {5}{2}}} \] Input:

integrate((d*x+c)**2/x**6/(b*x**2+a)**(1/2),x)
 

Output:

-3*a**4*b**(9/2)*c**2*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b* 
*5*x**6 + 15*a**3*b**6*x**8) - 2*a**3*b**(11/2)*c**2*x**2*sqrt(a/(b*x**2) 
+ 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 3*a**2* 
b**(13/2)*c**2*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5 
*x**6 + 15*a**3*b**6*x**8) - 12*a*b**(15/2)*c**2*x**6*sqrt(a/(b*x**2) + 1) 
/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 8*b**(17/2) 
*c**2*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 1 
5*a**3*b**6*x**8) - c*d/(2*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + sqrt(b)*c* 
d/(4*a*x**3*sqrt(a/(b*x**2) + 1)) - sqrt(b)*d**2*sqrt(a/(b*x**2) + 1)/(3*a 
*x**2) + 3*b**(3/2)*c*d/(4*a**2*x*sqrt(a/(b*x**2) + 1)) + 2*b**(3/2)*d**2* 
sqrt(a/(b*x**2) + 1)/(3*a**2) - 3*b**2*c*d*asinh(sqrt(a)/(sqrt(b)*x))/(4*a 
**(5/2))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=-\frac {3 \, b^{2} c d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{4 \, a^{\frac {5}{2}}} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} c^{2}}{15 \, a^{3} x} + \frac {2 \, \sqrt {b x^{2} + a} b d^{2}}{3 \, a^{2} x} + \frac {3 \, \sqrt {b x^{2} + a} b c d}{4 \, a^{2} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a} b c^{2}}{15 \, a^{2} x^{3}} - \frac {\sqrt {b x^{2} + a} d^{2}}{3 \, a x^{3}} - \frac {\sqrt {b x^{2} + a} c d}{2 \, a x^{4}} - \frac {\sqrt {b x^{2} + a} c^{2}}{5 \, a x^{5}} \] Input:

integrate((d*x+c)^2/x^6/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

-3/4*b^2*c*d*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 8/15*sqrt(b*x^2 + a)* 
b^2*c^2/(a^3*x) + 2/3*sqrt(b*x^2 + a)*b*d^2/(a^2*x) + 3/4*sqrt(b*x^2 + a)* 
b*c*d/(a^2*x^2) + 4/15*sqrt(b*x^2 + a)*b*c^2/(a^2*x^3) - 1/3*sqrt(b*x^2 + 
a)*d^2/(a*x^3) - 1/2*sqrt(b*x^2 + a)*c*d/(a*x^4) - 1/5*sqrt(b*x^2 + a)*c^2 
/(a*x^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (145) = 290\).

Time = 0.14 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.04 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=\frac {3 \, b^{2} c d \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{2}} - \frac {45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} b^{2} c d - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a b^{2} c d - 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {3}{2}} d^{2} - 320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} c^{2} + 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {3}{2}} d^{2} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{3} b^{2} c d + 160 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {5}{2}} c^{2} - 200 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {3}{2}} d^{2} - 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{4} b^{2} c d - 32 \, a^{4} b^{\frac {5}{2}} c^{2} + 40 \, a^{5} b^{\frac {3}{2}} d^{2}}{30 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{2}} \] Input:

integrate((d*x+c)^2/x^6/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

3/2*b^2*c*d*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) 
 - 1/30*(45*(sqrt(b)*x - sqrt(b*x^2 + a))^9*b^2*c*d - 210*(sqrt(b)*x - sqr 
t(b*x^2 + a))^7*a*b^2*c*d - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(3/2 
)*d^2 - 320*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2)*c^2 + 280*(sqrt(b) 
*x - sqrt(b*x^2 + a))^4*a^3*b^(3/2)*d^2 + 210*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^3*a^3*b^2*c*d + 160*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(5/2)*c^2 - 20 
0*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(3/2)*d^2 - 45*(sqrt(b)*x - sqrt(b 
*x^2 + a))*a^4*b^2*c*d - 32*a^4*b^(5/2)*c^2 + 40*a^5*b^(3/2)*d^2)/(((sqrt( 
b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{x^6\,\sqrt {b\,x^2+a}} \,d x \] Input:

int((c + d*x)^2/(x^6*(a + b*x^2)^(1/2)),x)
 

Output:

int((c + d*x)^2/(x^6*(a + b*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.30 \[ \int \frac {(c+d x)^2}{x^6 \sqrt {a+b x^2}} \, dx=\frac {-12 \sqrt {b \,x^{2}+a}\, a^{2} c^{2}-30 \sqrt {b \,x^{2}+a}\, a^{2} c d x -20 \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x^{2}+16 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{2}+45 \sqrt {b \,x^{2}+a}\, a b c d \,x^{3}+40 \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{4}-32 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x^{4}+45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{5}-40 \sqrt {b}\, a b \,d^{2} x^{5}+32 \sqrt {b}\, b^{2} c^{2} x^{5}}{60 a^{3} x^{5}} \] Input:

int((d*x+c)^2/x^6/(b*x^2+a)^(1/2),x)
 

Output:

( - 12*sqrt(a + b*x**2)*a**2*c**2 - 30*sqrt(a + b*x**2)*a**2*c*d*x - 20*sq 
rt(a + b*x**2)*a**2*d**2*x**2 + 16*sqrt(a + b*x**2)*a*b*c**2*x**2 + 45*sqr 
t(a + b*x**2)*a*b*c*d*x**3 + 40*sqrt(a + b*x**2)*a*b*d**2*x**4 - 32*sqrt(a 
 + b*x**2)*b**2*c**2*x**4 + 45*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + s 
qrt(b)*x)/sqrt(a))*b**2*c*d*x**5 - 45*sqrt(a)*log((sqrt(a + b*x**2) + sqrt 
(a) + sqrt(b)*x)/sqrt(a))*b**2*c*d*x**5 - 40*sqrt(b)*a*b*d**2*x**5 + 32*sq 
rt(b)*b**2*c**2*x**5)/(60*a**3*x**5)