\(\int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx\) [994]

Optimal result

Integrand size = 22, antiderivative size = 181 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=\frac {\left (b c^2-2 a d^2\right ) \sqrt {a+b x^2}}{8 a x^4}+\frac {b \left (b c^2-2 a d^2\right ) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac {2 c d \left (a+b x^2\right )^{3/2}}{5 a x^5}+\frac {4 b c d \left (a+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {b^2 \left (b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}} \] Output:

1/8*(-2*a*d^2+b*c^2)*(b*x^2+a)^(1/2)/a/x^4+1/16*b*(-2*a*d^2+b*c^2)*(b*x^2+ 
a)^(1/2)/a^2/x^2-1/6*c^2*(b*x^2+a)^(3/2)/a/x^6-2/5*c*d*(b*x^2+a)^(3/2)/a/x 
^5+4/15*b*c*d*(b*x^2+a)^(3/2)/a^2/x^3-1/16*b^2*(-2*a*d^2+b*c^2)*arctanh((b 
*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=\frac {\sqrt {a+b x^2} \left (b^2 c x^4 (15 c+64 d x)-2 a b x^2 \left (5 c^2+16 c d x+15 d^2 x^2\right )-4 a^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )\right )}{240 a^2 x^6}+\frac {b^2 \left (b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(b^2*c*x^4*(15*c + 64*d*x) - 2*a*b*x^2*(5*c^2 + 16*c*d*x 
+ 15*d^2*x^2) - 4*a^2*(10*c^2 + 24*c*d*x + 15*d^2*x^2)))/(240*a^2*x^6) + ( 
b^2*(b*c^2 - 2*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(8*a 
^(5/2))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {540, 27, 539, 27, 539, 25, 27, 534, 243, 51, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} (c+d x)^2}{x^7} \, dx\)

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 539

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{5} \int \frac {\left (5 \left (b c^2-2 a d^2\right )+8 b c d x\right ) \sqrt {b x^2+a}}{x^5}dx-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{5} \left (\frac {\int -\frac {b \left (32 a c d-5 \left (b c^2-2 a d^2\right ) x\right ) \sqrt {b x^2+a}}{x^4}dx}{4 a}+\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {\int \frac {b \left (32 a c d-5 \left (b c^2-2 a d^2\right ) x\right ) \sqrt {b x^2+a}}{x^4}dx}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {b \int \frac {\left (32 a c d-5 \left (b c^2-2 a d^2\right ) x\right ) \sqrt {b x^2+a}}{x^4}dx}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {b \left (-5 \left (b c^2-2 a d^2\right ) \int \frac {\sqrt {b x^2+a}}{x^3}dx-\frac {32 c d \left (a+b x^2\right )^{3/2}}{3 x^3}\right )}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {b \left (-\frac {5}{2} \left (b c^2-2 a d^2\right ) \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {32 c d \left (a+b x^2\right )^{3/2}}{3 x^3}\right )}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {b \left (-\frac {5}{2} \left (b c^2-2 a d^2\right ) \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {32 c d \left (a+b x^2\right )^{3/2}}{3 x^3}\right )}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {b \left (-\frac {5}{2} \left (b c^2-2 a d^2\right ) \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {32 c d \left (a+b x^2\right )^{3/2}}{3 x^3}\right )}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{5} \left (\frac {5 \left (a+b x^2\right )^{3/2} \left (b c^2-2 a d^2\right )}{4 a x^4}-\frac {b \left (-\frac {5}{2} \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right ) \left (b c^2-2 a d^2\right )-\frac {32 c d \left (a+b x^2\right )^{3/2}}{3 x^3}\right )}{4 a}\right )-\frac {4 c d \left (a+b x^2\right )^{3/2}}{5 x^5}}{2 a}-\frac {c^2 \left (a+b x^2\right )^{3/2}}{6 a x^6}\)

Input:

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

Output:

-1/6*(c^2*(a + b*x^2)^(3/2))/(a*x^6) + ((-4*c*d*(a + b*x^2)^(3/2))/(5*x^5) 
 + ((5*(b*c^2 - 2*a*d^2)*(a + b*x^2)^(3/2))/(4*a*x^4) - (b*((-32*c*d*(a + 
b*x^2)^(3/2))/(3*x^3) - (5*(b*c^2 - 2*a*d^2)*(-(Sqrt[a + b*x^2]/x^2) - (b* 
ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2))/(4*a))/5)/(2*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 51

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

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 = 0.43 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-64 b^{2} c d \,x^{5}+30 a b \,d^{2} x^{4}-15 b^{2} c^{2} x^{4}+32 a b c d \,x^{3}+60 a^{2} d^{2} x^{2}+10 a b \,c^{2} x^{2}+96 a^{2} c d x +40 a^{2} c^{2}\right )}{240 x^{6} a^{2}}+\frac {\left (2 a \,d^{2}-b \,c^{2}\right ) b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {5}{2}}}\)	\(141\)
default	\(c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )}{2 a}\right )+d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )+2 c d \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )\)	\(246\)

Input:

int((d*x+c)^2*(b*x^2+a)^(1/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/240*(b*x^2+a)^(1/2)*(-64*b^2*c*d*x^5+30*a*b*d^2*x^4-15*b^2*c^2*x^4+32*a 
*b*c*d*x^3+60*a^2*d^2*x^2+10*a*b*c^2*x^2+96*a^2*c*d*x+40*a^2*c^2)/x^6/a^2+ 
1/16*(2*a*d^2-b*c^2)*b^2/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.70 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=\left [-\frac {15 \, {\left (b^{3} c^{2} - 2 \, a b^{2} d^{2}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (64 \, a b^{2} c d x^{5} - 32 \, a^{2} b c d x^{3} - 96 \, a^{3} c d x - 40 \, a^{3} c^{2} + 15 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b d^{2}\right )} x^{4} - 10 \, {\left (a^{2} b c^{2} + 6 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{480 \, a^{3} x^{6}}, \frac {15 \, {\left (b^{3} c^{2} - 2 \, a b^{2} d^{2}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (64 \, a b^{2} c d x^{5} - 32 \, a^{2} b c d x^{3} - 96 \, a^{3} c d x - 40 \, a^{3} c^{2} + 15 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b d^{2}\right )} x^{4} - 10 \, {\left (a^{2} b c^{2} + 6 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{240 \, a^{3} x^{6}}\right ] \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(1/2)/x^7,x, algorithm="fricas")
 

Output:

[-1/480*(15*(b^3*c^2 - 2*a*b^2*d^2)*sqrt(a)*x^6*log(-(b*x^2 + 2*sqrt(b*x^2 
 + a)*sqrt(a) + 2*a)/x^2) - 2*(64*a*b^2*c*d*x^5 - 32*a^2*b*c*d*x^3 - 96*a^ 
3*c*d*x - 40*a^3*c^2 + 15*(a*b^2*c^2 - 2*a^2*b*d^2)*x^4 - 10*(a^2*b*c^2 + 
6*a^3*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3*x^6), 1/240*(15*(b^3*c^2 - 2*a*b^2*d 
^2)*sqrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (64*a*b^2*c*d*x^5 - 
32*a^2*b*c*d*x^3 - 96*a^3*c*d*x - 40*a^3*c^2 + 15*(a*b^2*c^2 - 2*a^2*b*d^2 
)*x^4 - 10*(a^2*b*c^2 + 6*a^3*d^2)*x^2)*sqrt(b*x^2 + a))/(a^3*x^6)]
 

Sympy [A] (verification not implemented)

Time = 7.57 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=- \frac {a c^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a d^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 \sqrt {b} c^{2}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b} c d \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {3 \sqrt {b} d^{2}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {3}{2}} c^{2}}{48 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 b^{\frac {3}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{15 a x^{2}} - \frac {b^{\frac {3}{2}} d^{2}}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b^{\frac {5}{2}} c^{2}}{16 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {4 b^{\frac {5}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{2}} + \frac {b^{2} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} - \frac {b^{3} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {5}{2}}} \] Input:

integrate((d*x+c)**2*(b*x**2+a)**(1/2)/x**7,x)
 

Output:

-a*c**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - a*d**2/(4*sqrt(b)*x**5*sqr 
t(a/(b*x**2) + 1)) - 5*sqrt(b)*c**2/(24*x**5*sqrt(a/(b*x**2) + 1)) - 2*sqr 
t(b)*c*d*sqrt(a/(b*x**2) + 1)/(5*x**4) - 3*sqrt(b)*d**2/(8*x**3*sqrt(a/(b* 
x**2) + 1)) + b**(3/2)*c**2/(48*a*x**3*sqrt(a/(b*x**2) + 1)) - 2*b**(3/2)* 
c*d*sqrt(a/(b*x**2) + 1)/(15*a*x**2) - b**(3/2)*d**2/(8*a*x*sqrt(a/(b*x**2 
) + 1)) + b**(5/2)*c**2/(16*a**2*x*sqrt(a/(b*x**2) + 1)) + 4*b**(5/2)*c*d* 
sqrt(a/(b*x**2) + 1)/(15*a**2) + b**2*d**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*a 
**(3/2)) - b**3*c**2*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(5/2))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=-\frac {b^{3} c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {5}{2}}} + \frac {b^{2} d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a} b^{3} c^{2}}{16 \, a^{3}} - \frac {\sqrt {b x^{2} + a} b^{2} d^{2}}{8 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c^{2}}{16 \, a^{3} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b d^{2}}{8 \, a^{2} x^{2}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b c d}{15 \, a^{2} x^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{2}}{8 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d^{2}}{4 \, a x^{4}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c d}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{2}}{6 \, a x^{6}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(1/2)/x^7,x, algorithm="maxima")
 

Output:

-1/16*b^3*c^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/8*b^2*d^2*arcsinh( 
a/(sqrt(a*b)*abs(x)))/a^(3/2) + 1/16*sqrt(b*x^2 + a)*b^3*c^2/a^3 - 1/8*sqr 
t(b*x^2 + a)*b^2*d^2/a^2 - 1/16*(b*x^2 + a)^(3/2)*b^2*c^2/(a^3*x^2) + 1/8* 
(b*x^2 + a)^(3/2)*b*d^2/(a^2*x^2) + 4/15*(b*x^2 + a)^(3/2)*b*c*d/(a^2*x^3) 
 + 1/8*(b*x^2 + a)^(3/2)*b*c^2/(a^2*x^4) - 1/4*(b*x^2 + a)^(3/2)*d^2/(a*x^ 
4) - 2/5*(b*x^2 + a)^(3/2)*c*d/(a*x^5) - 1/6*(b*x^2 + a)^(3/2)*c^2/(a*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (153) = 306\).

Time = 0.13 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.93 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{2}} - \frac {15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} b^{3} c^{2} - 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} a b^{2} d^{2} - 85 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} a b^{3} c^{2} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} a^{2} b^{2} d^{2} - 960 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} c d - 570 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a^{2} b^{3} c^{2} + 180 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a^{3} b^{2} d^{2} + 640 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c d - 570 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{3} b^{3} c^{2} + 180 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{4} b^{2} d^{2} - 85 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{4} b^{3} c^{2} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{5} b^{2} d^{2} + 384 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c d + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{5} b^{3} c^{2} - 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{6} b^{2} d^{2} - 64 \, a^{6} b^{\frac {5}{2}} c d}{120 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{6} a^{2}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(1/2)/x^7,x, algorithm="giac")
 

Output:

1/8*(b^3*c^2 - 2*a*b^2*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a) 
)/(sqrt(-a)*a^2) - 1/120*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^11*b^3*c^2 - 30 
*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a*b^2*d^2 - 85*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^9*a*b^3*c^2 - 150*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a^2*b^2*d^2 - 960*( 
sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*c*d - 570*(sqrt(b)*x - sqrt(b*x 
^2 + a))^7*a^2*b^3*c^2 + 180*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^3*b^2*d^2 + 
 640*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(5/2)*c*d - 570*(sqrt(b)*x - sq 
rt(b*x^2 + a))^5*a^3*b^3*c^2 + 180*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a^4*b^2 
*d^2 - 85*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^4*b^3*c^2 - 150*(sqrt(b)*x - s 
qrt(b*x^2 + a))^3*a^5*b^2*d^2 + 384*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^ 
(5/2)*c*d + 15*(sqrt(b)*x - sqrt(b*x^2 + a))*a^5*b^3*c^2 - 30*(sqrt(b)*x - 
 sqrt(b*x^2 + a))*a^6*b^2*d^2 - 64*a^6*b^(5/2)*c*d)/(((sqrt(b)*x - sqrt(b* 
x^2 + a))^2 - a)^6*a^2)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^(1/2)*(c + d*x)^2)/x^7,x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x)^2)/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.73 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2}}{x^7} \, dx=\frac {-40 \sqrt {b \,x^{2}+a}\, a^{3} c^{2}-96 \sqrt {b \,x^{2}+a}\, a^{3} c d x -60 \sqrt {b \,x^{2}+a}\, a^{3} d^{2} x^{2}-10 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} x^{2}-32 \sqrt {b \,x^{2}+a}\, a^{2} b c d \,x^{3}-30 \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{2} x^{4}+15 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x^{4}+64 \sqrt {b \,x^{2}+a}\, a \,b^{2} c d \,x^{5}-30 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d^{2} x^{6}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{6}+30 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d^{2} x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{6}-64 \sqrt {b}\, a \,b^{2} c d \,x^{6}}{240 a^{3} x^{6}} \] Input:

int((d*x+c)^2*(b*x^2+a)^(1/2)/x^7,x)
 

Output:

( - 40*sqrt(a + b*x**2)*a**3*c**2 - 96*sqrt(a + b*x**2)*a**3*c*d*x - 60*sq 
rt(a + b*x**2)*a**3*d**2*x**2 - 10*sqrt(a + b*x**2)*a**2*b*c**2*x**2 - 32* 
sqrt(a + b*x**2)*a**2*b*c*d*x**3 - 30*sqrt(a + b*x**2)*a**2*b*d**2*x**4 + 
15*sqrt(a + b*x**2)*a*b**2*c**2*x**4 + 64*sqrt(a + b*x**2)*a*b**2*c*d*x**5 
 - 30*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2 
*d**2*x**6 + 15*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt( 
a))*b**3*c**2*x**6 + 30*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)* 
x)/sqrt(a))*a*b**2*d**2*x**6 - 15*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) 
+ sqrt(b)*x)/sqrt(a))*b**3*c**2*x**6 - 64*sqrt(b)*a*b**2*c*d*x**6)/(240*a* 
*3*x**6)