\(\int \frac {(c+d x)^3 (a+b x^2)^{3/2}}{x^3} \, dx\) [1067]

Optimal result

Integrand size = 22, antiderivative size = 225 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx=\frac {3}{8} \left (4 c \left (b c^2+2 a d^2\right )+d \left (12 b c^2+a d^2\right ) x\right ) \sqrt {a+b x^2}+\frac {\left (2 c \left (b c^2+2 a d^2\right )+d \left (12 b c^2+a d^2\right ) x\right ) \left (a+b x^2\right )^{3/2}}{4 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {3 c^2 d \left (a+b x^2\right )^{5/2}}{a x}+\frac {3 a d \left (12 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}-\frac {3}{2} \sqrt {a} c \left (b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \] Output:

3/8*(4*c*(2*a*d^2+b*c^2)+d*(a*d^2+12*b*c^2)*x)*(b*x^2+a)^(1/2)+1/4*(2*c*(2 
*a*d^2+b*c^2)+d*(a*d^2+12*b*c^2)*x)*(b*x^2+a)^(3/2)/a-1/2*c^3*(b*x^2+a)^(5 
/2)/a/x^2-3*c^2*d*(b*x^2+a)^(5/2)/a/x+3/8*a*d*(a*d^2+12*b*c^2)*arctanh(b^( 
1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)-3/2*a^(1/2)*c*(2*a*d^2+b*c^2)*arctanh((b*x 
^2+a)^(1/2)/a^(1/2))
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx=3 \sqrt {a} c \left (b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {1}{8} \left (\frac {\sqrt {a+b x^2} \left (2 b x^2 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+a \left (-4 c^3-24 c^2 d x+32 c d^2 x^2+5 d^3 x^3\right )\right )}{x^2}-\frac {3 a d \left (12 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}\right ) \] Input:

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

Output:

3*Sqrt[a]*c*(b*c^2 + 2*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a 
]] + ((Sqrt[a + b*x^2]*(2*b*x^2*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3 
) + a*(-4*c^3 - 24*c^2*d*x + 32*c*d^2*x^2 + 5*d^3*x^3)))/x^2 - (3*a*d*(12* 
b*c^2 + a*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/8
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {540, 25, 2338, 25, 27, 535, 27, 535, 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.

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int -\frac {a \left (3 c \left (b c^2+2 a d^2\right )+2 d \left (12 b c^2+a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x}dx}{a}-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {a \left (3 c \left (b c^2+2 a d^2\right )+2 d \left (12 b c^2+a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x}dx}{a}-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (3 c \left (b c^2+2 a d^2\right )+2 d \left (12 b c^2+a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x}dx-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 535

\(\displaystyle \frac {\frac {1}{4} a \int \frac {6 \left (2 c \left (b c^2+2 a d^2\right )+d \left (12 b c^2+a d^2\right ) x\right ) \sqrt {b x^2+a}}{x}dx+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3}{2} a \int \frac {\left (2 c \left (b c^2+2 a d^2\right )+d \left (12 b c^2+a d^2\right ) x\right ) \sqrt {b x^2+a}}{x}dx+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 535

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \int \frac {4 c \left (b c^2+2 a d^2\right )+d \left (12 b c^2+a d^2\right ) x}{x \sqrt {b x^2+a}}dx+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \left (d \left (a d^2+12 b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx+4 c \left (2 a d^2+b c^2\right ) \int \frac {1}{x \sqrt {b x^2+a}}dx\right )+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \left (4 c \left (2 a d^2+b c^2\right ) \int \frac {1}{x \sqrt {b x^2+a}}dx+d \left (a d^2+12 b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}\right )+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \left (4 c \left (2 a d^2+b c^2\right ) \int \frac {1}{x \sqrt {b x^2+a}}dx+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \left (2 c \left (2 a d^2+b c^2\right ) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \left (\frac {4 c \left (2 a d^2+b c^2\right ) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3}{2} a \left (\frac {1}{2} a \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b}}-\frac {4 c \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (2 a d^2+b c^2\right )}{\sqrt {a}}\right )+\frac {1}{2} \sqrt {a+b x^2} \left (d x \left (a d^2+12 b c^2\right )+4 c \left (2 a d^2+b c^2\right )\right )\right )+\frac {1}{2} \left (a+b x^2\right )^{3/2} \left (d x \left (a d^2+12 b c^2\right )+2 c \left (2 a d^2+b c^2\right )\right )-\frac {6 c^2 d \left (a+b x^2\right )^{5/2}}{x}}{2 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{2 a x^2}\)

Input:

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

Output:

-1/2*(c^3*(a + b*x^2)^(5/2))/(a*x^2) + (((2*c*(b*c^2 + 2*a*d^2) + d*(12*b* 
c^2 + a*d^2)*x)*(a + b*x^2)^(3/2))/2 - (6*c^2*d*(a + b*x^2)^(5/2))/x + (3* 
a*(((4*c*(b*c^2 + 2*a*d^2) + d*(12*b*c^2 + a*d^2)*x)*Sqrt[a + b*x^2])/2 + 
(a*((d*(12*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - 
(4*c*(b*c^2 + 2*a*d^2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2))/2)/ 
(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 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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 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]
 

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 = 242, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {a \,c^{2} \sqrt {b \,x^{2}+a}\, \left (6 d x +c \right )}{2 x^{2}}+\frac {3 a^{2} d^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 \sqrt {b}}+c^{3} b \sqrt {b \,x^{2}+a}+\frac {b \,d^{3} x^{3} \sqrt {b \,x^{2}+a}}{4}+\frac {5 d^{3} a x \sqrt {b \,x^{2}+a}}{8}+\frac {3 b \,c^{2} d x \sqrt {b \,x^{2}+a}}{2}+\frac {9 a \sqrt {b}\, c^{2} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}-3 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a^{\frac {3}{2}} c \,d^{2}-\frac {3 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) \sqrt {a}\, b \,c^{3}}{2}+b c \,d^{2} x^{2} \sqrt {b \,x^{2}+a}+4 a c \,d^{2} \sqrt {b \,x^{2}+a}\)	\(242\)
default	\(d^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+c^{3} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )+3 c \,d^{2} \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )+3 c^{2} d \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )\)	\(276\)

Input:

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

Output:

-1/2*a*c^2*(b*x^2+a)^(1/2)*(6*d*x+c)/x^2+3/8*a^2*d^3*ln(b^(1/2)*x+(b*x^2+a 
)^(1/2))/b^(1/2)+c^3*b*(b*x^2+a)^(1/2)+1/4*b*d^3*x^3*(b*x^2+a)^(1/2)+5/8*d 
^3*a*x*(b*x^2+a)^(1/2)+3/2*b*c^2*d*x*(b*x^2+a)^(1/2)+9/2*a*b^(1/2)*c^2*d*l 
n(b^(1/2)*x+(b*x^2+a)^(1/2))-3*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)*a^(3/ 
2)*c*d^2-3/2*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)*a^(1/2)*b*c^3+b*c*d^2*x 
^2*(b*x^2+a)^(1/2)+4*a*c*d^2*(b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 791, normalized size of antiderivative = 3.52 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/16*(3*(12*a*b*c^2*d + a^2*d^3)*sqrt(b)*x^2*log(-2*b*x^2 - 2*sqrt(b*x^2 
+ a)*sqrt(b)*x - a) + 12*(b^2*c^3 + 2*a*b*c*d^2)*sqrt(a)*x^2*log(-(b*x^2 - 
 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(2*b^2*d^3*x^5 + 8*b^2*c*d^2*x^ 
4 - 24*a*b*c^2*d*x - 4*a*b*c^3 + (12*b^2*c^2*d + 5*a*b*d^3)*x^3 + 8*(b^2*c 
^3 + 4*a*b*c*d^2)*x^2)*sqrt(b*x^2 + a))/(b*x^2), -1/8*(3*(12*a*b*c^2*d + a 
^2*d^3)*sqrt(-b)*x^2*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 6*(b^2*c^3 + 2*a 
*b*c*d^2)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) 
- (2*b^2*d^3*x^5 + 8*b^2*c*d^2*x^4 - 24*a*b*c^2*d*x - 4*a*b*c^3 + (12*b^2* 
c^2*d + 5*a*b*d^3)*x^3 + 8*(b^2*c^3 + 4*a*b*c*d^2)*x^2)*sqrt(b*x^2 + a))/( 
b*x^2), 1/16*(24*(b^2*c^3 + 2*a*b*c*d^2)*sqrt(-a)*x^2*arctan(sqrt(b*x^2 + 
a)*sqrt(-a)/a) + 3*(12*a*b*c^2*d + a^2*d^3)*sqrt(b)*x^2*log(-2*b*x^2 - 2*s 
qrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(2*b^2*d^3*x^5 + 8*b^2*c*d^2*x^4 - 24*a* 
b*c^2*d*x - 4*a*b*c^3 + (12*b^2*c^2*d + 5*a*b*d^3)*x^3 + 8*(b^2*c^3 + 4*a* 
b*c*d^2)*x^2)*sqrt(b*x^2 + a))/(b*x^2), -1/8*(3*(12*a*b*c^2*d + a^2*d^3)*s 
qrt(-b)*x^2*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 12*(b^2*c^3 + 2*a*b*c*d^2 
)*sqrt(-a)*x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (2*b^2*d^3*x^5 + 8*b^2 
*c*d^2*x^4 - 24*a*b*c^2*d*x - 4*a*b*c^3 + (12*b^2*c^2*d + 5*a*b*d^3)*x^3 + 
 8*(b^2*c^3 + 4*a*b*c*d^2)*x^2)*sqrt(b*x^2 + a))/(b*x^2)]
 

Sympy [A] (verification not implemented)

Time = 4.51 (sec) , antiderivative size = 551, normalized size of antiderivative = 2.45 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-3*a**(3/2)*c**2*d/(x*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*c*d**2*asinh(sqrt(a 
)/(sqrt(b)*x)) - 3*sqrt(a)*b*c**3*asinh(sqrt(a)/(sqrt(b)*x))/2 - 3*sqrt(a) 
*b*c**2*d*x/sqrt(1 + b*x**2/a) + 3*a**2*c*d**2/(sqrt(b)*x*sqrt(a/(b*x**2) 
+ 1)) - a*sqrt(b)*c**3*sqrt(a/(b*x**2) + 1)/(2*x) + a*sqrt(b)*c**3/(x*sqrt 
(a/(b*x**2) + 1)) + 3*a*sqrt(b)*c**2*d*asinh(sqrt(b)*x/sqrt(a)) + 3*a*sqrt 
(b)*c*d**2*x/sqrt(a/(b*x**2) + 1) + a*d**3*Piecewise((a*Piecewise((log(2*s 
qrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2 
), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True)) + b**(3/ 
2)*c**3*x/sqrt(a/(b*x**2) + 1) + 3*b*c**2*d*Piecewise((a*Piecewise((log(2* 
sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x** 
2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True)) + 3*b*c 
*d**2*Piecewise((a*sqrt(a + b*x**2)/(3*b) + x**2*sqrt(a + b*x**2)/3, Ne(b, 
 0)), (sqrt(a)*x**2/2, True)) + b*d**3*Piecewise((-a**2*Piecewise((log(2*s 
qrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2 
), True))/(8*b) + a*x*sqrt(a + b*x**2)/(8*b) + x**3*sqrt(a + b*x**2)/4, Ne 
(b, 0)), (sqrt(a)*x**3/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx=\frac {9}{2} \, \sqrt {b x^{2} + a} b c^{2} d x + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} d^{3} x + \frac {3}{8} \, \sqrt {b x^{2} + a} a d^{3} x + \frac {9}{2} \, a \sqrt {b} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + \frac {3 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} - \frac {3}{2} \, \sqrt {a} b c^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - 3 \, a^{\frac {3}{2}} c d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b x^{2} + a} b c^{3} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{3}}{2 \, a} + {\left (b x^{2} + a\right )}^{\frac {3}{2}} c d^{2} + 3 \, \sqrt {b x^{2} + a} a c d^{2} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{2} d}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{3}}{2 \, a x^{2}} \] Input:

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

Output:

9/2*sqrt(b*x^2 + a)*b*c^2*d*x + 1/4*(b*x^2 + a)^(3/2)*d^3*x + 3/8*sqrt(b*x 
^2 + a)*a*d^3*x + 9/2*a*sqrt(b)*c^2*d*arcsinh(b*x/sqrt(a*b)) + 3/8*a^2*d^3 
*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 3/2*sqrt(a)*b*c^3*arcsinh(a/(sqrt(a*b)*a 
bs(x))) - 3*a^(3/2)*c*d^2*arcsinh(a/(sqrt(a*b)*abs(x))) + 3/2*sqrt(b*x^2 + 
 a)*b*c^3 + 1/2*(b*x^2 + a)^(3/2)*b*c^3/a + (b*x^2 + a)^(3/2)*c*d^2 + 3*sq 
rt(b*x^2 + a)*a*c*d^2 - 3*(b*x^2 + a)^(3/2)*c^2*d/x - 1/2*(b*x^2 + a)^(5/2 
)*c^3/(a*x^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx=\frac {1}{8} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left (b d^{3} x + 4 \, b c d^{2}\right )} x + \frac {12 \, b^{3} c^{2} d + 5 \, a b^{2} d^{3}}{b^{2}}\right )} x + \frac {8 \, {\left (b^{3} c^{3} + 4 \, a b^{2} c d^{2}\right )}}{b^{2}}\right )} + \frac {3 \, {\left (a b c^{3} + 2 \, a^{2} c d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3 \, {\left (12 \, a b c^{2} d + a^{2} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, \sqrt {b}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c^{3} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} c^{2} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c^{3} - 6 \, a^{3} \sqrt {b} c^{2} d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \] Input:

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

Output:

1/8*sqrt(b*x^2 + a)*((2*(b*d^3*x + 4*b*c*d^2)*x + (12*b^3*c^2*d + 5*a*b^2* 
d^3)/b^2)*x + 8*(b^3*c^3 + 4*a*b^2*c*d^2)/b^2) + 3*(a*b*c^3 + 2*a^2*c*d^2) 
*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - 3/8*(12*a*b*c^ 
2*d + a^2*d^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + ((sqrt(b)* 
x - sqrt(b*x^2 + a))^3*a*b*c^3 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqr 
t(b)*c^2*d + (sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b*c^3 - 6*a^3*sqrt(b)*c^2*d 
)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^3} \, dx=\frac {-4 \sqrt {b \,x^{2}+a}\, a b \,c^{3}-24 \sqrt {b \,x^{2}+a}\, a b \,c^{2} d x +32 \sqrt {b \,x^{2}+a}\, a b c \,d^{2} x^{2}+5 \sqrt {b \,x^{2}+a}\, a b \,d^{3} x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{2} c^{3} x^{2}+12 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} d \,x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{2} c \,d^{2} x^{4}+2 \sqrt {b \,x^{2}+a}\, b^{2} d^{3} x^{5}+24 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,d^{2} x^{2}+12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{3} x^{2}-24 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,d^{2} x^{2}-12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{3} x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d^{3} x^{2}+36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} d \,x^{2}}{8 b \,x^{2}} \] Input:

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

Output:

( - 4*sqrt(a + b*x**2)*a*b*c**3 - 24*sqrt(a + b*x**2)*a*b*c**2*d*x + 32*sq 
rt(a + b*x**2)*a*b*c*d**2*x**2 + 5*sqrt(a + b*x**2)*a*b*d**3*x**3 + 8*sqrt 
(a + b*x**2)*b**2*c**3*x**2 + 12*sqrt(a + b*x**2)*b**2*c**2*d*x**3 + 8*sqr 
t(a + b*x**2)*b**2*c*d**2*x**4 + 2*sqrt(a + b*x**2)*b**2*d**3*x**5 + 24*sq 
rt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*d**2*x** 
2 + 12*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2* 
c**3*x**2 - 24*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a 
))*a*b*c*d**2*x**2 - 12*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)* 
x)/sqrt(a))*b**2*c**3*x**2 + 3*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/ 
sqrt(a))*a**2*d**3*x**2 + 36*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sq 
rt(a))*a*b*c**2*d*x**2)/(8*b*x**2)