\(\int \frac {(c+d x)^2 (a+b x^2)^{3/2}}{x^6} \, dx\) [1056]

Optimal result

Integrand size = 22, antiderivative size = 140 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=-\frac {b d (3 c+4 d x) \sqrt {a+b x^2}}{4 x^2}-\frac {d (3 c+2 d x) \left (a+b x^2\right )^{3/2}}{6 x^4}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {3 b^2 c d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {a+b x^2} \left (12 b^2 c^2 x^4+2 a^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )+a b x^2 \left (24 c^2+75 c d x+80 d^2 x^2\right )\right )}{60 a x^5}+\frac {3 b^2 c d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}}-b^{3/2} d^2 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((c + d*x)^2*(a + b*x^2)^(3/2))/x^6,x]
 

Output:

-1/60*(Sqrt[a + b*x^2]*(12*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*(24*c^2 + 75*c*d*x + 80*d^2*x^2)))/(a*x^5) + (3*b^2*c*d*Arc 
Tanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(2*Sqrt[a]) - b^(3/2)*d^2*Log 
[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {540, 27, 537, 27, 537, 25, 538, 224, 219, 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.

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)^2*(a + b*x^2)^(3/2))/x^6,x]
 

Output:

-1/5*(c^2*(a + b*x^2)^(5/2))/(a*x^5) + d*(-1/6*((3*c + 2*d*x)*(a + b*x^2)^ 
(3/2))/x^4 + (b*(-1/2*((3*c + 4*d*x)*Sqrt[a + b*x^2])/x^2 + (b*((4*d*ArcTa 
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (3*c*ArcTanh[Sqrt[a + b*x^2]/Sq 
rt[a]])/Sqrt[a]))/2))/2)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 537

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

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

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 = 0.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (80 a b \,d^{2} x^{4}+12 b^{2} c^{2} x^{4}+75 a b c d \,x^{3}+20 a^{2} d^{2} x^{2}+24 a b \,c^{2} x^{2}+30 a^{2} c d x +12 a^{2} c^{2}\right )}{60 x^{5} a}+b^{\frac {3}{2}} d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-\frac {3 b^{2} d c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{4 \sqrt {a}}\)	\(143\)
default	\(-\frac {c^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}+d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )+2 c d \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )\)	\(229\)

Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/60*(b*x^2+a)^(1/2)*(80*a*b*d^2*x^4+12*b^2*c^2*x^4+75*a*b*c*d*x^3+20*a^2 
*d^2*x^2+24*a*b*c^2*x^2+30*a^2*c*d*x+12*a^2*c^2)/x^5/a+b^(3/2)*d^2*ln(b^(1 
/2)*x+(b*x^2+a)^(1/2))-3/4*b^2*d*c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/ 
2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 644, normalized size of antiderivative = 4.60 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=\left [\frac {60 \, a b^{\frac {3}{2}} d^{2} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 45 \, \sqrt {a} b^{2} c d x^{5} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (75 \, a b c d x^{3} + 30 \, a^{2} c d x + 4 \, {\left (3 \, b^{2} c^{2} + 20 \, a b d^{2}\right )} x^{4} + 12 \, a^{2} c^{2} + 4 \, {\left (6 \, a b c^{2} + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{120 \, a x^{5}}, -\frac {120 \, a \sqrt {-b} b d^{2} x^{5} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 45 \, \sqrt {a} b^{2} c d x^{5} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (75 \, a b c d x^{3} + 30 \, a^{2} c d x + 4 \, {\left (3 \, b^{2} c^{2} + 20 \, a b d^{2}\right )} x^{4} + 12 \, a^{2} c^{2} + 4 \, {\left (6 \, a b c^{2} + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{120 \, a x^{5}}, \frac {45 \, \sqrt {-a} b^{2} c d x^{5} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + 30 \, a b^{\frac {3}{2}} d^{2} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (75 \, a b c d x^{3} + 30 \, a^{2} c d x + 4 \, {\left (3 \, b^{2} c^{2} + 20 \, a b d^{2}\right )} x^{4} + 12 \, a^{2} c^{2} + 4 \, {\left (6 \, a b c^{2} + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{60 \, a x^{5}}, -\frac {60 \, a \sqrt {-b} b d^{2} x^{5} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 45 \, \sqrt {-a} b^{2} c d x^{5} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (75 \, a b c d x^{3} + 30 \, a^{2} c d x + 4 \, {\left (3 \, b^{2} c^{2} + 20 \, a b d^{2}\right )} x^{4} + 12 \, a^{2} c^{2} + 4 \, {\left (6 \, a b c^{2} + 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{60 \, a x^{5}}\right ] \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^6,x, algorithm="fricas")
 

Output:

[1/120*(60*a*b^(3/2)*d^2*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - 
a) + 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*(75*a*b*c*d*x^3 + 30*a^2*c*d*x + 4*(3*b^2*c^2 + 20*a*b*d^2)*x^4 
+ 12*a^2*c^2 + 4*(6*a*b*c^2 + 5*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a*x^5), -1 
/120*(120*a*sqrt(-b)*b*d^2*x^5*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 45*sqr 
t(a)*b^2*c*d*x^5*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*( 
75*a*b*c*d*x^3 + 30*a^2*c*d*x + 4*(3*b^2*c^2 + 20*a*b*d^2)*x^4 + 12*a^2*c^ 
2 + 4*(6*a*b*c^2 + 5*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a*x^5), 1/60*(45*sqrt 
(-a)*b^2*c*d*x^5*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 30*a*b^(3/2)*d^2*x^5 
*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - (75*a*b*c*d*x^3 + 30*a^ 
2*c*d*x + 4*(3*b^2*c^2 + 20*a*b*d^2)*x^4 + 12*a^2*c^2 + 4*(6*a*b*c^2 + 5*a 
^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a*x^5), -1/60*(60*a*sqrt(-b)*b*d^2*x^5*arct 
an(sqrt(-b)*x/sqrt(b*x^2 + a)) - 45*sqrt(-a)*b^2*c*d*x^5*arctan(sqrt(b*x^2 
 + a)*sqrt(-a)/a) + (75*a*b*c*d*x^3 + 30*a^2*c*d*x + 4*(3*b^2*c^2 + 20*a*b 
*d^2)*x^4 + 12*a^2*c^2 + 4*(6*a*b*c^2 + 5*a^2*d^2)*x^2)*sqrt(b*x^2 + a))/( 
a*x^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (131) = 262\).

Time = 5.83 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.36 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=- \frac {\sqrt {a} b d^{2}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{2} c d}{2 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {3 a \sqrt {b} c d}{4 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a \sqrt {b} d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {2 b^{\frac {3}{2}} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {b^{\frac {3}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{x} - \frac {b^{\frac {3}{2}} c d}{4 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {b^{\frac {3}{2}} d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + b^{\frac {3}{2}} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {b^{\frac {5}{2}} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {3 b^{2} c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{4 \sqrt {a}} - \frac {b^{2} d^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

integrate((d*x+c)**2*(b*x**2+a)**(3/2)/x**6,x)
 

Output:

-sqrt(a)*b*d**2/(x*sqrt(1 + b*x**2/a)) - a**2*c*d/(2*sqrt(b)*x**5*sqrt(a/( 
b*x**2) + 1)) - a*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/(5*x**4) - 3*a*sqrt(b) 
*c*d/(4*x**3*sqrt(a/(b*x**2) + 1)) - a*sqrt(b)*d**2*sqrt(a/(b*x**2) + 1)/( 
3*x**2) - 2*b**(3/2)*c**2*sqrt(a/(b*x**2) + 1)/(5*x**2) - b**(3/2)*c*d*sqr 
t(a/(b*x**2) + 1)/x - b**(3/2)*c*d/(4*x*sqrt(a/(b*x**2) + 1)) - b**(3/2)*d 
**2*sqrt(a/(b*x**2) + 1)/3 + b**(3/2)*d**2*asinh(sqrt(b)*x/sqrt(a)) - b**( 
5/2)*c**2*sqrt(a/(b*x**2) + 1)/(5*a) - 3*b**2*c*d*asinh(sqrt(a)/(sqrt(b)*x 
))/(4*sqrt(a)) - b**2*d**2*x/(sqrt(a)*sqrt(1 + b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.41 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=\frac {\sqrt {b x^{2} + a} b^{2} d^{2} x}{a} + b^{\frac {3}{2}} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3 \, b^{2} c d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{4 \, \sqrt {a}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c d}{4 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b^{2} c d}{4 \, a} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b d^{2}}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b c d}{4 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{2}}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c d}{2 \, a x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2}}{5 \, a x^{5}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^6,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (114) = 228\).

Time = 0.15 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.84 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=\frac {3 \, b^{2} c d \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} - b^{\frac {3}{2}} d^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} b^{2} c d + 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {5}{2}} c^{2} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {3}{2}} d^{2} - 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a b^{2} c d - 360 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {3}{2}} d^{2} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} c^{2} + 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {3}{2}} d^{2} + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{3} b^{2} c d - 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {3}{2}} d^{2} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{4} b^{2} c d + 12 \, a^{4} b^{\frac {5}{2}} c^{2} + 80 \, a^{5} b^{\frac {3}{2}} d^{2}}{30 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^6,x, algorithm="giac")
 

Output:

3/2*b^2*c*d*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - b^( 
3/2)*d^2*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) + 1/30*(75*(sqrt(b)*x - sq 
rt(b*x^2 + a))^9*b^2*c*d + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(5/2)*c^2 
+ 120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(3/2)*d^2 - 30*(sqrt(b)*x - sqrt 
(b*x^2 + a))^7*a*b^2*c*d - 360*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(3/2) 
*d^2 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2)*c^2 + 440*(sqrt(b)* 
x - sqrt(b*x^2 + a))^4*a^3*b^(3/2)*d^2 + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
3*a^3*b^2*c*d - 280*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(3/2)*d^2 - 75*( 
sqrt(b)*x - sqrt(b*x^2 + a))*a^4*b^2*c*d + 12*a^4*b^(5/2)*c^2 + 80*a^5*b^( 
3/2)*d^2)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2}{x^6} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)^2)/x^6,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)^2)/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.83 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx=\frac {-12 \sqrt {b \,x^{2}+a}\, a^{2} c^{2}-30 \sqrt {b \,x^{2}+a}\, a^{2} c d x -20 \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x^{2}-24 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{2}-75 \sqrt {b \,x^{2}+a}\, a b c d \,x^{3}-80 \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{4}-12 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x^{4}+45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{5}+60 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{5}+32 \sqrt {b}\, a b \,d^{2} x^{5}-12 \sqrt {b}\, b^{2} c^{2} x^{5}}{60 a \,x^{5}} \] Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)/x^6,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 - 24*sqrt(a + b*x**2)*a*b*c**2*x**2 - 75*sqr 
t(a + b*x**2)*a*b*c*d*x**3 - 80*sqrt(a + b*x**2)*a*b*d**2*x**4 - 12*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 + 60*sqrt(b)*log((sqrt(a + b*x**2) 
 + sqrt(b)*x)/sqrt(a))*a*b*d**2*x**5 + 32*sqrt(b)*a*b*d**2*x**5 - 12*sqrt( 
b)*b**2*c**2*x**5)/(60*a*x**5)