\(\int \frac {(c+d x) (a+b x^2)^{3/2} (A+B x+C x^2)}{x^{10}} \, dx\) [60]

Optimal result

Integrand size = 30, antiderivative size = 288 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=-\frac {b (8 a C d-3 b (B c+A d)) \sqrt {a+b x^2}}{64 a x^4}-\frac {b^2 (8 a C d-3 b (B c+A d)) \sqrt {a+b x^2}}{128 a^2 x^2}-\frac {(8 a C d-3 b (B c+A d)) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}-\frac {(B c+A d) \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {(4 A b c-9 a (c C+B d)) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b (4 A b c-9 a (c C+B d)) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}+\frac {b^3 (8 a C d-3 b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \] Output:

-1/64*b*(8*a*C*d-3*b*(A*d+B*c))*(b*x^2+a)^(1/2)/a/x^4-1/128*b^2*(8*a*C*d-3 
*b*(A*d+B*c))*(b*x^2+a)^(1/2)/a^2/x^2-1/48*(8*a*C*d-3*b*(A*d+B*c))*(b*x^2+ 
a)^(3/2)/a/x^6-1/9*A*c*(b*x^2+a)^(5/2)/a/x^9-1/8*(A*d+B*c)*(b*x^2+a)^(5/2) 
/a/x^8+1/63*(4*A*b*c-9*a*(B*d+C*c))*(b*x^2+a)^(5/2)/a^2/x^7-2/315*b*(4*A*b 
*c-9*a*(B*d+C*c))*(b*x^2+a)^(5/2)/a^3/x^5+1/128*b^3*(8*a*C*d-3*b*(A*d+B*c) 
)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 3.62 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=-\frac {\sqrt {a+b x^2} \left (1024 A b^4 c x^8-a b^3 x^6 (A (512 c+945 d x)+9 x (105 B c+256 c C x+256 B d x))+6 a^2 b^2 x^4 \left (A (64 c+105 d x)+3 x \left (35 B c+64 c C x+64 B d x+140 C d x^2\right )\right )+8 a^3 b x^2 \left (A (800 c+945 d x)+3 x \left (315 B c+384 c C x+384 B d x+490 C d x^2\right )\right )+80 a^4 (7 A (8 c+9 d x)+3 x (4 C x (6 c+7 d x)+3 B (7 c+8 d x)))\right )}{40320 a^3 x^9}-\frac {b^3 C d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}-\frac {3 b^4 (B c+A d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{64 a^{5/2}} \] Input:

Integrate[((c + d*x)*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^10,x]
 

Output:

-1/40320*(Sqrt[a + b*x^2]*(1024*A*b^4*c*x^8 - a*b^3*x^6*(A*(512*c + 945*d* 
x) + 9*x*(105*B*c + 256*c*C*x + 256*B*d*x)) + 6*a^2*b^2*x^4*(A*(64*c + 105 
*d*x) + 3*x*(35*B*c + 64*c*C*x + 64*B*d*x + 140*C*d*x^2)) + 8*a^3*b*x^2*(A 
*(800*c + 945*d*x) + 3*x*(315*B*c + 384*c*C*x + 384*B*d*x + 490*C*d*x^2)) 
+ 80*a^4*(7*A*(8*c + 9*d*x) + 3*x*(4*C*x*(6*c + 7*d*x) + 3*B*(7*c + 8*d*x) 
))))/(a^3*x^9) - (b^3*C*d*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/ 
(8*a^(3/2)) - (3*b^4*(B*c + A*d)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/ 
Sqrt[a]])/(64*a^(5/2))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2338, 25, 2338, 27, 539, 539, 27, 534, 243, 51, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} (c+d x) \left (A+B x+C x^2\right )}{x^{10}} \, dx\)

\(\Big \downarrow \) 2338

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (9 a C d x^2-(4 A b c-9 a (c C+B d)) x+9 a (B c+A d)\right )}{x^9}dx}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (9 a C d x^2-(4 A b c-9 a (c C+B d)) x+9 a (B c+A d)\right )}{x^9}dx}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int \frac {a (8 (4 A b c-9 a (c C+B d))-9 (8 a C d-3 b (B c+A d)) x) \left (b x^2+a\right )^{3/2}}{x^8}dx}{8 a}-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{8} \int \frac {(8 (4 A b c-9 a (c C+B d))-9 (8 a C d-3 b (B c+A d)) x) \left (b x^2+a\right )^{3/2}}{x^8}dx-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\int \frac {(63 a (8 a C d-3 b (B c+A d))+16 b (4 A b c-9 a (c C+B d)) x) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{8} \left (\frac {-\frac {\int -\frac {3 a b (32 (4 A b c-9 a (c C+B d))-21 (8 a C d-3 b (B c+A d)) x) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \int \frac {(32 (4 A b c-9 a (c C+B d))-21 (8 a C d-3 b (B c+A d)) x) \left (b x^2+a\right )^{3/2}}{x^6}dx-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \left (-21 (8 a C d-3 b (A d+B c)) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \left (-\frac {21}{2} (8 a C d-3 b (A d+B c)) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \left (-\frac {21}{2} (8 a C d-3 b (A d+B c)) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \left (-\frac {21}{2} (8 a C d-3 b (A d+B c)) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \left (-\frac {21}{2} (8 a C d-3 b (A d+B c)) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{8} \left (\frac {\frac {1}{2} b \left (-\frac {21}{2} \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right ) (8 a C d-3 b (A d+B c))-\frac {32 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{5 a x^5}\right )-\frac {21 \left (a+b x^2\right )^{5/2} (8 a C d-3 b (A d+B c))}{2 x^6}}{7 a}+\frac {8 \left (a+b x^2\right )^{5/2} (4 A b c-9 a (B d+c C))}{7 a x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} (A d+B c)}{8 x^8}}{9 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{9 a x^9}\)

Input:

Int[((c + d*x)*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^10,x]
 

Output:

-1/9*(A*c*(a + b*x^2)^(5/2))/(a*x^9) + ((-9*(B*c + A*d)*(a + b*x^2)^(5/2)) 
/(8*x^8) + ((8*(4*A*b*c - 9*a*(c*C + B*d))*(a + b*x^2)^(5/2))/(7*a*x^7) + 
((-21*(8*a*C*d - 3*b*(B*c + A*d))*(a + b*x^2)^(5/2))/(2*x^6) + (b*((-32*(4 
*A*b*c - 9*a*(c*C + B*d))*(a + b*x^2)^(5/2))/(5*a*x^5) - (21*(8*a*C*d - 3* 
b*(B*c + A*d))*(-1/2*(a + b*x^2)^(3/2)/x^4 + (3*b*(-(Sqrt[a + b*x^2]/x^2) 
- (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/4))/2))/2)/(7*a))/8)/(9*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 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 = 0.71 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (1024 A \,b^{4} c \,x^{8}-2304 B a \,b^{3} d \,x^{8}-2304 C a \,b^{3} c \,x^{8}-945 A a \,b^{3} d \,x^{7}-945 B a \,b^{3} c \,x^{7}+2520 C \,a^{2} b^{2} d \,x^{7}-512 A a \,b^{3} c \,x^{6}+1152 B \,a^{2} b^{2} d \,x^{6}+1152 C \,a^{2} b^{2} c \,x^{6}+630 A \,a^{2} b^{2} d \,x^{5}+630 B \,a^{2} b^{2} c \,x^{5}+11760 C \,a^{3} b d \,x^{5}+384 A \,a^{2} b^{2} c \,x^{4}+9216 B \,a^{3} b d \,x^{4}+9216 C \,a^{3} b c \,x^{4}+7560 A \,a^{3} b d \,x^{3}+7560 B \,a^{3} b c \,x^{3}+6720 C \,a^{4} d \,x^{3}+6400 A \,a^{3} b c \,x^{2}+5760 B \,a^{4} d \,x^{2}+5760 C \,a^{4} c \,x^{2}+5040 A \,a^{4} d x +5040 B \,a^{4} c x +4480 A \,a^{4} c \right )}{40320 x^{9} a^{3}}-\frac {\left (3 A b d +3 B b c -8 a C d \right ) b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {5}{2}}}\)	\(329\)
default	\(\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )+A c \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )+d C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )\)	\(392\)

Input:

int((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^10,x,method=_RETURNVERBOSE)
 

Output:

-1/40320*(b*x^2+a)^(1/2)*(1024*A*b^4*c*x^8-2304*B*a*b^3*d*x^8-2304*C*a*b^3 
*c*x^8-945*A*a*b^3*d*x^7-945*B*a*b^3*c*x^7+2520*C*a^2*b^2*d*x^7-512*A*a*b^ 
3*c*x^6+1152*B*a^2*b^2*d*x^6+1152*C*a^2*b^2*c*x^6+630*A*a^2*b^2*d*x^5+630* 
B*a^2*b^2*c*x^5+11760*C*a^3*b*d*x^5+384*A*a^2*b^2*c*x^4+9216*B*a^3*b*d*x^4 
+9216*C*a^3*b*c*x^4+7560*A*a^3*b*d*x^3+7560*B*a^3*b*c*x^3+6720*C*a^4*d*x^3 
+6400*A*a^3*b*c*x^2+5760*B*a^4*d*x^2+5760*C*a^4*c*x^2+5040*A*a^4*d*x+5040* 
B*a^4*c*x+4480*A*a^4*c)/x^9/a^3-1/128*(3*A*b*d+3*B*b*c-8*C*a*d)/a^(5/2)*b^ 
3*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.27 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^10,x, algorithm="fricas" 
)
 

Output:

[1/80640*(315*(3*B*b^4*c - (8*C*a*b^3 - 3*A*b^4)*d)*sqrt(a)*x^9*log(-(b*x^ 
2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(256*(9*B*a*b^3*d + (9*C*a*b 
^3 - 4*A*b^4)*c)*x^8 + 315*(3*B*a*b^3*c - (8*C*a^2*b^2 - 3*A*a*b^3)*d)*x^7 
 - 128*(9*B*a^2*b^2*d + (9*C*a^2*b^2 - 4*A*a*b^3)*c)*x^6 - 4480*A*a^4*c - 
210*(3*B*a^2*b^2*c + (56*C*a^3*b + 3*A*a^2*b^2)*d)*x^5 - 384*(24*B*a^3*b*d 
 + (24*C*a^3*b + A*a^2*b^2)*c)*x^4 - 840*(9*B*a^3*b*c + (8*C*a^4 + 9*A*a^3 
*b)*d)*x^3 - 640*(9*B*a^4*d + (9*C*a^4 + 10*A*a^3*b)*c)*x^2 - 5040*(B*a^4* 
c + A*a^4*d)*x)*sqrt(b*x^2 + a))/(a^3*x^9), 1/40320*(315*(3*B*b^4*c - (8*C 
*a*b^3 - 3*A*b^4)*d)*sqrt(-a)*x^9*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (25 
6*(9*B*a*b^3*d + (9*C*a*b^3 - 4*A*b^4)*c)*x^8 + 315*(3*B*a*b^3*c - (8*C*a^ 
2*b^2 - 3*A*a*b^3)*d)*x^7 - 128*(9*B*a^2*b^2*d + (9*C*a^2*b^2 - 4*A*a*b^3) 
*c)*x^6 - 4480*A*a^4*c - 210*(3*B*a^2*b^2*c + (56*C*a^3*b + 3*A*a^2*b^2)*d 
)*x^5 - 384*(24*B*a^3*b*d + (24*C*a^3*b + A*a^2*b^2)*c)*x^4 - 840*(9*B*a^3 
*b*c + (8*C*a^4 + 9*A*a^3*b)*d)*x^3 - 640*(9*B*a^4*d + (9*C*a^4 + 10*A*a^3 
*b)*c)*x^2 - 5040*(B*a^4*c + A*a^4*d)*x)*sqrt(b*x^2 + a))/(a^3*x^9)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2387 vs. \(2 (270) = 540\).

Time = 56.08 (sec) , antiderivative size = 2387, normalized size of antiderivative = 8.29 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**10,x)
 

Output:

-35*A*a**8*b**(19/2)*c*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6 
*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 110*A*a**7*b 
**(21/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10* 
x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*A*a**6*b**(23/2 
)*c*x**4*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 
 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 40*A*a**5*b**(25/2)*c*x**6 
*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a** 
5*b**11*x**12 + 315*a**4*b**12*x**14) - 15*A*a**5*b**(11/2)*c*sqrt(a/(b*x* 
*2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 
 5*A*a**4*b**(27/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945* 
a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 33*A*a** 
4*b**(13/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b** 
5*x**8 + 105*a**3*b**6*x**10) + 30*A*a**3*b**(29/2)*c*x**10*sqrt(a/(b*x**2 
) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 
 315*a**4*b**12*x**14) - 17*A*a**3*b**(15/2)*c*x**4*sqrt(a/(b*x**2) + 1)/( 
105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 40*A*a**2 
*b**(31/2)*c*x**12*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b** 
10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 3*A*a**2*b**(17/ 
2)*c*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 
105*a**3*b**6*x**10) - A*a**2*d/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) +...
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=\frac {C b^{3} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b^{3} d}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} C b^{3} d}{16 \, a^{2}} - \frac {3 \, {\left (B c + A d\right )} b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b^{4}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} {\left (B c + A d\right )} b^{4}}{128 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C b^{2} d}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} b^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C b d}{24 \, a^{2} x^{4}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2} c}{315 \, a^{3} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} b^{2}}{64 \, a^{3} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d}{6 \, a x^{6}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} b}{35 \, a^{2} x^{5}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b c}{63 \, a^{2} x^{7}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} b}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )}}{7 \, a x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c}{9 \, a x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )}}{8 \, a x^{8}} \] Input:

integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^10,x, algorithm="maxima" 
)
 

Output:

1/16*C*b^3*d*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 1/48*(b*x^2 + a)^(3/2 
)*C*b^3*d/a^3 - 1/16*sqrt(b*x^2 + a)*C*b^3*d/a^2 - 3/128*(B*c + A*d)*b^4*a 
rcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/128*(b*x^2 + a)^(3/2)*(B*c + A*d) 
*b^4/a^4 + 3/128*sqrt(b*x^2 + a)*(B*c + A*d)*b^4/a^3 + 1/48*(b*x^2 + a)^(5 
/2)*C*b^2*d/(a^3*x^2) - 1/128*(b*x^2 + a)^(5/2)*(B*c + A*d)*b^3/(a^4*x^2) 
+ 1/24*(b*x^2 + a)^(5/2)*C*b*d/(a^2*x^4) - 8/315*(b*x^2 + a)^(5/2)*A*b^2*c 
/(a^3*x^5) - 1/64*(b*x^2 + a)^(5/2)*(B*c + A*d)*b^2/(a^3*x^4) - 1/6*(b*x^2 
 + a)^(5/2)*C*d/(a*x^6) + 2/35*(b*x^2 + a)^(5/2)*(C*c + B*d)*b/(a^2*x^5) + 
 4/63*(b*x^2 + a)^(5/2)*A*b*c/(a^2*x^7) + 1/16*(b*x^2 + a)^(5/2)*(B*c + A* 
d)*b/(a^2*x^6) - 1/7*(b*x^2 + a)^(5/2)*(C*c + B*d)/(a*x^7) - 1/9*(b*x^2 + 
a)^(5/2)*A*c/(a*x^9) - 1/8*(b*x^2 + a)^(5/2)*(B*c + A*d)/(a*x^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1378 vs. \(2 (252) = 504\).

Time = 0.18 (sec) , antiderivative size = 1378, normalized size of antiderivative = 4.78 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^10,x, algorithm="giac")
 

Output:

1/64*(3*B*b^4*c - 8*C*a*b^3*d + 3*A*b^4*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 
 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/20160*(945*(sqrt(b)*x - sqrt(b*x^2 + a 
))^17*B*b^4*c - 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^17*C*a*b^3*d + 945*(sqr 
t(b)*x - sqrt(b*x^2 + a))^17*A*b^4*d - 8190*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
15*B*a*b^4*c - 31920*(sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a^2*b^3*d - 8190*( 
sqrt(b)*x - sqrt(b*x^2 + a))^15*A*a*b^4*d - 80640*(sqrt(b)*x - sqrt(b*x^2 
+ a))^14*C*a^2*b^(7/2)*c - 80640*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^2*b^ 
(7/2)*d - 97650*(sqrt(b)*x - sqrt(b*x^2 + a))^13*B*a^2*b^4*c + 45360*(sqrt 
(b)*x - sqrt(b*x^2 + a))^13*C*a^3*b^3*d - 97650*(sqrt(b)*x - sqrt(b*x^2 + 
a))^13*A*a^2*b^4*d + 80640*(sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^3*b^(7/2)* 
c - 215040*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*a^2*b^(9/2)*c + 80640*(sqrt( 
b)*x - sqrt(b*x^2 + a))^12*B*a^3*b^(7/2)*d - 106470*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^11*B*a^3*b^4*c + 15120*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a^4*b^3* 
d - 106470*(sqrt(b)*x - sqrt(b*x^2 + a))^11*A*a^3*b^4*d - 80640*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^10*C*a^4*b^(7/2)*c - 322560*(sqrt(b)*x - sqrt(b*x^2 + 
a))^10*A*a^3*b^(9/2)*c - 80640*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^4*b^(7 
/2)*d + 209664*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^5*b^(7/2)*c - 451584*(s 
qrt(b)*x - sqrt(b*x^2 + a))^8*A*a^4*b^(9/2)*c + 209664*(sqrt(b)*x - sqrt(b 
*x^2 + a))^8*B*a^5*b^(7/2)*d + 106470*(sqrt(b)*x - sqrt(b*x^2 + a))^7*B*a^ 
5*b^4*c - 15120*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^6*b^3*d + 106470*(s...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^{10}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2))/x^10,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2))/x^10, x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.44 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{10}} \, dx=\frac {-2520 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c d \,x^{9}+2520 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c d \,x^{9}-6400 \sqrt {b \,x^{2}+a}\, a^{4} b c \,x^{2}-5040 \sqrt {b \,x^{2}+a}\, a^{4} b c x -7560 \sqrt {b \,x^{2}+a}\, a^{4} b d \,x^{3}-5760 \sqrt {b \,x^{2}+a}\, a^{4} b d \,x^{2}-6720 \sqrt {b \,x^{2}+a}\, a^{4} c d \,x^{3}-384 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c \,x^{4}-7560 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c \,x^{3}-630 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d \,x^{5}-9216 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d \,x^{4}-9216 \sqrt {b \,x^{2}+a}\, a^{3} b \,c^{2} x^{4}+512 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,x^{6}-630 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,x^{5}+945 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{7}-1152 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{6}-1152 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} x^{6}-1024 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{8}+945 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{7}+2304 \sqrt {b \,x^{2}+a}\, a \,b^{4} d \,x^{8}+2304 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} x^{8}+945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} c \,x^{9}-945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} c \,x^{9}+1024 \sqrt {b}\, a \,b^{4} c \,x^{9}-2304 \sqrt {b}\, a \,b^{4} d \,x^{9}-2304 \sqrt {b}\, a \,b^{3} c^{2} x^{9}-4480 \sqrt {b \,x^{2}+a}\, a^{5} c -11760 \sqrt {b \,x^{2}+a}\, a^{3} b c d \,x^{5}-2520 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c d \,x^{7}+945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} d \,x^{9}-945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} d \,x^{9}-5040 \sqrt {b \,x^{2}+a}\, a^{5} d x -5760 \sqrt {b \,x^{2}+a}\, a^{4} c^{2} x^{2}}{40320 a^{3} x^{9}} \] Input:

int((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^10,x)
 

Output:

( - 4480*sqrt(a + b*x**2)*a**5*c - 5040*sqrt(a + b*x**2)*a**5*d*x - 6400*s 
qrt(a + b*x**2)*a**4*b*c*x**2 - 5040*sqrt(a + b*x**2)*a**4*b*c*x - 7560*sq 
rt(a + b*x**2)*a**4*b*d*x**3 - 5760*sqrt(a + b*x**2)*a**4*b*d*x**2 - 5760* 
sqrt(a + b*x**2)*a**4*c**2*x**2 - 6720*sqrt(a + b*x**2)*a**4*c*d*x**3 - 38 
4*sqrt(a + b*x**2)*a**3*b**2*c*x**4 - 7560*sqrt(a + b*x**2)*a**3*b**2*c*x* 
*3 - 630*sqrt(a + b*x**2)*a**3*b**2*d*x**5 - 9216*sqrt(a + b*x**2)*a**3*b* 
*2*d*x**4 - 9216*sqrt(a + b*x**2)*a**3*b*c**2*x**4 - 11760*sqrt(a + b*x**2 
)*a**3*b*c*d*x**5 + 512*sqrt(a + b*x**2)*a**2*b**3*c*x**6 - 630*sqrt(a + b 
*x**2)*a**2*b**3*c*x**5 + 945*sqrt(a + b*x**2)*a**2*b**3*d*x**7 - 1152*sqr 
t(a + b*x**2)*a**2*b**3*d*x**6 - 1152*sqrt(a + b*x**2)*a**2*b**2*c**2*x**6 
 - 2520*sqrt(a + b*x**2)*a**2*b**2*c*d*x**7 - 1024*sqrt(a + b*x**2)*a*b**4 
*c*x**8 + 945*sqrt(a + b*x**2)*a*b**4*c*x**7 + 2304*sqrt(a + b*x**2)*a*b** 
4*d*x**8 + 2304*sqrt(a + b*x**2)*a*b**3*c**2*x**8 + 945*sqrt(a)*log((sqrt( 
a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**4*d*x**9 - 2520*sqrt(a)*l 
og((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*c*d*x**9 + 945 
*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**5*c*x**9 
 - 945*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b** 
4*d*x**9 + 2520*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt( 
a))*a*b**3*c*d*x**9 - 945*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b 
)*x)/sqrt(a))*b**5*c*x**9 + 1024*sqrt(b)*a*b**4*c*x**9 - 2304*sqrt(b)*a...