\(\int \frac {(c+d x)^3 (a+b x^2)^{3/2}}{x^7} \, dx\) [1071]

Optimal result

Integrand size = 22, antiderivative size = 208 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\frac {b \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right ) \sqrt {a+b x^2}}{16 a x^2}+\frac {\left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right ) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {3 c^2 d \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {b^2 c \left (b c^2-18 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:

1/16*b*(c*(-18*a*d^2+b*c^2)-16*a*d^3*x)*(b*x^2+a)^(1/2)/a/x^2+1/24*(c*(-18 
*a*d^2+b*c^2)-8*a*d^3*x)*(b*x^2+a)^(3/2)/a/x^4-1/6*c^3*(b*x^2+a)^(5/2)/a/x 
^6-3/5*c^2*d*(b*x^2+a)^(5/2)/a/x^5+b^(3/2)*d^3*arctanh(b^(1/2)*x/(b*x^2+a) 
^(1/2))+1/16*b^2*c*(-18*a*d^2+b*c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3 
/2)
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {\sqrt {a+b x^2} \left (3 b^2 c^2 x^4 (5 c+48 d x)+4 a^2 \left (10 c^3+36 c^2 d x+45 c d^2 x^2+20 d^3 x^3\right )+2 a b x^2 \left (35 c^3+144 c^2 d x+225 c d^2 x^2+160 d^3 x^3\right )\right )}{240 a x^6}+\frac {b^2 c \left (-b c^2+18 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-b^{3/2} d^3 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((c + d*x)^3*(a + b*x^2)^(3/2))/x^7,x]
 

Output:

-1/240*(Sqrt[a + b*x^2]*(3*b^2*c^2*x^4*(5*c + 48*d*x) + 4*a^2*(10*c^3 + 36 
*c^2*d*x + 45*c*d^2*x^2 + 20*d^3*x^3) + 2*a*b*x^2*(35*c^3 + 144*c^2*d*x + 
225*c*d^2*x^2 + 160*d^3*x^3)))/(a*x^6) + (b^2*c*(-(b*c^2) + 18*a*d^2)*ArcT 
anh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(8*a^(3/2)) - b^(3/2)*d^3*Log[ 
-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.03, 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, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} (c+d x)^3}{x^7} \, dx\)

\(\Big \downarrow \) 540

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (6 a x^2 d^3+18 a c^2 d-c \left (b c^2-18 a d^2\right ) x\right )}{x^6}dx}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (6 a x^2 d^3+18 a c^2 d-c \left (b c^2-18 a d^2\right ) x\right )}{x^6}dx}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int \frac {5 a \left (c \left (b c^2-18 a d^2\right )-6 a d^3 x\right ) \left (b x^2+a\right )^{3/2}}{x^5}dx}{5 a}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\int \frac {\left (c \left (b c^2-18 a d^2\right )-6 a d^3 x\right ) \left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {\frac {1}{4} b \int -\frac {3 \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right ) \sqrt {b x^2+a}}{x^3}dx+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3}{4} b \int \frac {\left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right ) \sqrt {b x^2+a}}{x^3}dx+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 537

\(\displaystyle \frac {-\frac {3}{4} b \left (-\frac {1}{2} b \int -\frac {c \left (b c^2-18 a d^2\right )-16 a d^3 x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \int \frac {c \left (b c^2-18 a d^2\right )-16 a d^3 x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (c \left (b c^2-18 a d^2\right ) \int \frac {1}{x \sqrt {b x^2+a}}dx-16 a d^3 \int \frac {1}{\sqrt {b x^2+a}}dx\right )-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (c \left (b c^2-18 a d^2\right ) \int \frac {1}{x \sqrt {b x^2+a}}dx-16 a d^3 \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}\right )-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (c \left (b c^2-18 a d^2\right ) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {16 a d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {1}{2} c \left (b c^2-18 a d^2\right ) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {16 a d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (\frac {c \left (b c^2-18 a d^2\right ) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {16 a d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {3}{4} b \left (\frac {1}{2} b \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (b c^2-18 a d^2\right )}{\sqrt {a}}-\frac {16 a d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )-\frac {\sqrt {a+b x^2} \left (c \left (b c^2-18 a d^2\right )-16 a d^3 x\right )}{2 x^2}\right )+\frac {\left (a+b x^2\right )^{3/2} \left (c \left (b c^2-18 a d^2\right )-8 a d^3 x\right )}{4 x^4}-\frac {18 c^2 d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^3 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

Input:

Int[((c + d*x)^3*(a + b*x^2)^(3/2))/x^7,x]
 

Output:

-1/6*(c^3*(a + b*x^2)^(5/2))/(a*x^6) + (((c*(b*c^2 - 18*a*d^2) - 8*a*d^3*x 
)*(a + b*x^2)^(3/2))/(4*x^4) - (18*c^2*d*(a + b*x^2)^(5/2))/(5*x^5) - (3*b 
*(-1/2*((c*(b*c^2 - 18*a*d^2) - 16*a*d^3*x)*Sqrt[a + b*x^2])/x^2 + (b*((-1 
6*a*d^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (c*(b*c^2 - 18*a*d 
^2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2))/4)/(6*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 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]
 

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 = 0.48 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (320 a b \,d^{3} x^{5}+144 b^{2} c^{2} d \,x^{5}+450 a b c \,d^{2} x^{4}+15 b^{2} x^{4} c^{3}+80 a^{2} d^{3} x^{3}+288 a b d \,x^{3} c^{2}+180 a^{2} c \,d^{2} x^{2}+70 a b \,c^{3} x^{2}+144 a^{2} c^{2} d x +40 a^{2} c^{3}\right )}{240 x^{6} a}+\frac {b^{2} \left (-\frac {c \left (18 a \,d^{2}-b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+\frac {16 a \,d^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\right )}{16 a}\)	\(202\)
default	\(c^{3} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+d^{3} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )+3 c \,d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )-\frac {3 c^{2} d \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}\)	\(360\)

Input:

int((d*x+c)^3*(b*x^2+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/240*(b*x^2+a)^(1/2)*(320*a*b*d^3*x^5+144*b^2*c^2*d*x^5+450*a*b*c*d^2*x^ 
4+15*b^2*c^3*x^4+80*a^2*d^3*x^3+288*a*b*c^2*d*x^3+180*a^2*c*d^2*x^2+70*a*b 
*c^3*x^2+144*a^2*c^2*d*x+40*a^2*c^3)/x^6/a+1/16/a*b^2*(-c*(18*a*d^2-b*c^2) 
/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+16*a*d^3*ln(b^(1/2)*x+(b*x^ 
2+a)^(1/2))/b^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 896, normalized size of antiderivative = 4.31 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(3/2)/x^7,x, algorithm="fricas")
 

Output:

[1/480*(240*a^2*b^(3/2)*d^3*x^6*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x 
 - a) - 15*(b^3*c^3 - 18*a*b^2*c*d^2)*sqrt(a)*x^6*log(-(b*x^2 - 2*sqrt(b*x 
^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(144*a^3*c^2*d*x + 40*a^3*c^3 + 16*(9*a*b^ 
2*c^2*d + 20*a^2*b*d^3)*x^5 + 15*(a*b^2*c^3 + 30*a^2*b*c*d^2)*x^4 + 16*(18 
*a^2*b*c^2*d + 5*a^3*d^3)*x^3 + 10*(7*a^2*b*c^3 + 18*a^3*c*d^2)*x^2)*sqrt( 
b*x^2 + a))/(a^2*x^6), -1/480*(480*a^2*sqrt(-b)*b*d^3*x^6*arctan(sqrt(-b)* 
x/sqrt(b*x^2 + a)) + 15*(b^3*c^3 - 18*a*b^2*c*d^2)*sqrt(a)*x^6*log(-(b*x^2 
 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(144*a^3*c^2*d*x + 40*a^3*c^3 
 + 16*(9*a*b^2*c^2*d + 20*a^2*b*d^3)*x^5 + 15*(a*b^2*c^3 + 30*a^2*b*c*d^2) 
*x^4 + 16*(18*a^2*b*c^2*d + 5*a^3*d^3)*x^3 + 10*(7*a^2*b*c^3 + 18*a^3*c*d^ 
2)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6), 1/240*(120*a^2*b^(3/2)*d^3*x^6*log(-2* 
b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 15*(b^3*c^3 - 18*a*b^2*c*d^2)*s 
qrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (144*a^3*c^2*d*x + 40*a^3 
*c^3 + 16*(9*a*b^2*c^2*d + 20*a^2*b*d^3)*x^5 + 15*(a*b^2*c^3 + 30*a^2*b*c* 
d^2)*x^4 + 16*(18*a^2*b*c^2*d + 5*a^3*d^3)*x^3 + 10*(7*a^2*b*c^3 + 18*a^3* 
c*d^2)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6), -1/240*(240*a^2*sqrt(-b)*b*d^3*x^6 
*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + 15*(b^3*c^3 - 18*a*b^2*c*d^2)*sqrt(- 
a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (144*a^3*c^2*d*x + 40*a^3*c^3 
+ 16*(9*a*b^2*c^2*d + 20*a^2*b*d^3)*x^5 + 15*(a*b^2*c^3 + 30*a^2*b*c*d^2)* 
x^4 + 16*(18*a^2*b*c^2*d + 5*a^3*d^3)*x^3 + 10*(7*a^2*b*c^3 + 18*a^3*c*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (190) = 380\).

Time = 11.70 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.36 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=- \frac {\sqrt {a} b d^{3}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{2} c^{3}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 a^{2} c d^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 a \sqrt {b} c^{3}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 a \sqrt {b} c^{2} d \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {9 a \sqrt {b} c d^{2}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a \sqrt {b} d^{3} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {17 b^{\frac {3}{2}} c^{3}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {6 b^{\frac {3}{2}} c^{2} d \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {3 b^{\frac {3}{2}} c d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {3 b^{\frac {3}{2}} c d^{2}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {b^{\frac {3}{2}} d^{3} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + b^{\frac {3}{2}} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {b^{\frac {5}{2}} c^{3}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 b^{\frac {5}{2}} c^{2} d \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {9 b^{2} c d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} - \frac {b^{2} d^{3} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} \] Input:

integrate((d*x+c)**3*(b*x**2+a)**(3/2)/x**7,x)
 

Output:

-sqrt(a)*b*d**3/(x*sqrt(1 + b*x**2/a)) - a**2*c**3/(6*sqrt(b)*x**7*sqrt(a/ 
(b*x**2) + 1)) - 3*a**2*c*d**2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 11* 
a*sqrt(b)*c**3/(24*x**5*sqrt(a/(b*x**2) + 1)) - 3*a*sqrt(b)*c**2*d*sqrt(a/ 
(b*x**2) + 1)/(5*x**4) - 9*a*sqrt(b)*c*d**2/(8*x**3*sqrt(a/(b*x**2) + 1)) 
- a*sqrt(b)*d**3*sqrt(a/(b*x**2) + 1)/(3*x**2) - 17*b**(3/2)*c**3/(48*x**3 
*sqrt(a/(b*x**2) + 1)) - 6*b**(3/2)*c**2*d*sqrt(a/(b*x**2) + 1)/(5*x**2) - 
 3*b**(3/2)*c*d**2*sqrt(a/(b*x**2) + 1)/(2*x) - 3*b**(3/2)*c*d**2/(8*x*sqr 
t(a/(b*x**2) + 1)) - b**(3/2)*d**3*sqrt(a/(b*x**2) + 1)/3 + b**(3/2)*d**3* 
asinh(sqrt(b)*x/sqrt(a)) - b**(5/2)*c**3/(16*a*x*sqrt(a/(b*x**2) + 1)) - 3 
*b**(5/2)*c**2*d*sqrt(a/(b*x**2) + 1)/(5*a) - 9*b**2*c*d**2*asinh(sqrt(a)/ 
(sqrt(b)*x))/(8*sqrt(a)) - b**2*d**3*x/(sqrt(a)*sqrt(1 + b*x**2/a)) + b**3 
*c**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(3/2))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\frac {\sqrt {b x^{2} + a} b^{2} d^{3} x}{a} + b^{\frac {3}{2}} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + \frac {b^{3} c^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {9 \, b^{2} c d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} c^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} b^{3} c^{3}}{16 \, a^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} c d^{2}}{8 \, a^{2}} + \frac {9 \, \sqrt {b x^{2} + a} b^{2} c d^{2}}{8 \, a} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b d^{3}}{3 \, a x} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} c^{3}}{48 \, a^{3} x^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b c d^{2}}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{3}}{3 \, a x^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b c^{3}}{24 \, a^{2} x^{4}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d^{2}}{4 \, a x^{4}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} d}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{3}}{6 \, a x^{6}} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(3/2)/x^7,x, algorithm="maxima")
 

Output:

sqrt(b*x^2 + a)*b^2*d^3*x/a + b^(3/2)*d^3*arcsinh(b*x/sqrt(a*b)) + 1/16*b^ 
3*c^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 9/8*b^2*c*d^2*arcsinh(a/(sqr 
t(a*b)*abs(x)))/sqrt(a) - 1/48*(b*x^2 + a)^(3/2)*b^3*c^3/a^3 - 1/16*sqrt(b 
*x^2 + a)*b^3*c^3/a^2 + 3/8*(b*x^2 + a)^(3/2)*b^2*c*d^2/a^2 + 9/8*sqrt(b*x 
^2 + a)*b^2*c*d^2/a - 2/3*(b*x^2 + a)^(3/2)*b*d^3/(a*x) + 1/48*(b*x^2 + a) 
^(5/2)*b^2*c^3/(a^3*x^2) - 3/8*(b*x^2 + a)^(5/2)*b*c*d^2/(a^2*x^2) - 1/3*( 
b*x^2 + a)^(5/2)*d^3/(a*x^3) + 1/24*(b*x^2 + a)^(5/2)*b*c^3/(a^2*x^4) - 3/ 
4*(b*x^2 + a)^(5/2)*c*d^2/(a*x^4) - 3/5*(b*x^2 + a)^(5/2)*c^2*d/(a*x^5) - 
1/6*(b*x^2 + a)^(5/2)*c^3/(a*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 792 vs. \(2 (180) = 360\).

Time = 0.15 (sec) , antiderivative size = 792, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(3/2)/x^7,x, algorithm="giac")
 

Output:

-b^(3/2)*d^3*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) - 1/8*(b^3*c^3 - 18*a* 
b^2*c*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 
1/120*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^11*b^3*c^3 + 450*(sqrt(b)*x - sqrt 
(b*x^2 + a))^11*a*b^2*c*d^2 + 720*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/ 
2)*c^2*d + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*d^3 + 235*(sqr 
t(b)*x - sqrt(b*x^2 + a))^9*a*b^3*c^3 - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
9*a^2*b^2*c*d^2 - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*c^2*d - 
1920*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*d^3 + 390*(sqrt(b)*x - sq 
rt(b*x^2 + a))^7*a^2*b^3*c^3 + 180*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^3*b^2 
*c*d^2 + 1440*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(5/2)*c^2*d + 3200*(sq 
rt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(3/2)*d^3 + 390*(sqrt(b)*x - sqrt(b*x^2 
 + a))^5*a^3*b^3*c^3 + 180*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a^4*b^2*c*d^2 - 
 1440*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*c^2*d - 2880*(sqrt(b)*x 
- sqrt(b*x^2 + a))^4*a^5*b^(3/2)*d^3 + 235*(sqrt(b)*x - sqrt(b*x^2 + a))^3 
*a^4*b^3*c^3 - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^5*b^2*c*d^2 + 144*(sq 
rt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*c^2*d + 1440*(sqrt(b)*x - sqrt(b* 
x^2 + a))^2*a^6*b^(3/2)*d^3 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))*a^5*b^3*c^3 
 + 450*(sqrt(b)*x - sqrt(b*x^2 + a))*a^6*b^2*c*d^2 - 144*a^6*b^(5/2)*c^2*d 
 - 320*a^7*b^(3/2)*d^3)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^6*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^3}{x^7} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)^3)/x^7,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)^3)/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\frac {-40 \sqrt {b \,x^{2}+a}\, a^{3} c^{3}-144 \sqrt {b \,x^{2}+a}\, a^{3} c^{2} d x -180 \sqrt {b \,x^{2}+a}\, a^{3} c \,d^{2} x^{2}-80 \sqrt {b \,x^{2}+a}\, a^{3} d^{3} x^{3}-70 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{3} x^{2}-288 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} d \,x^{3}-450 \sqrt {b \,x^{2}+a}\, a^{2} b c \,d^{2} x^{4}-320 \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{3} x^{5}-15 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{3} x^{4}-144 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} d \,x^{5}+270 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c \,d^{2} x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{3} x^{6}-270 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c \,d^{2} x^{6}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{3} x^{6}+240 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b \,d^{3} x^{6}+160 \sqrt {b}\, a^{2} b \,d^{3} x^{6}-96 \sqrt {b}\, a \,b^{2} c^{2} d \,x^{6}}{240 a^{2} x^{6}} \] Input:

int((d*x+c)^3*(b*x^2+a)^(3/2)/x^7,x)
 

Output:

( - 40*sqrt(a + b*x**2)*a**3*c**3 - 144*sqrt(a + b*x**2)*a**3*c**2*d*x - 1 
80*sqrt(a + b*x**2)*a**3*c*d**2*x**2 - 80*sqrt(a + b*x**2)*a**3*d**3*x**3 
- 70*sqrt(a + b*x**2)*a**2*b*c**3*x**2 - 288*sqrt(a + b*x**2)*a**2*b*c**2* 
d*x**3 - 450*sqrt(a + b*x**2)*a**2*b*c*d**2*x**4 - 320*sqrt(a + b*x**2)*a* 
*2*b*d**3*x**5 - 15*sqrt(a + b*x**2)*a*b**2*c**3*x**4 - 144*sqrt(a + b*x** 
2)*a*b**2*c**2*d*x**5 + 270*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt 
(b)*x)/sqrt(a))*a*b**2*c*d**2*x**6 - 15*sqrt(a)*log((sqrt(a + b*x**2) - sq 
rt(a) + sqrt(b)*x)/sqrt(a))*b**3*c**3*x**6 - 270*sqrt(a)*log((sqrt(a + b*x 
**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*c*d**2*x**6 + 15*sqrt(a)*log(( 
sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c**3*x**6 + 240*sqrt 
(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*d**3*x**6 + 160*sqr 
t(b)*a**2*b*d**3*x**6 - 96*sqrt(b)*a*b**2*c**2*d*x**6)/(240*a**2*x**6)