\(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx\) [115]

Optimal result

Integrand size = 23, antiderivative size = 310 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}} \] Output:

1/512*(4*a*b*B*(-12*a*c+7*b^2)-A*(16*a^2*c^2-56*a*b^2*c+21*b^4))*(b*x+2*a) 
*(c*x^2+b*x+a)^(1/2)/a^5/x^2-1/6*A*(c*x^2+b*x+a)^(3/2)/a/x^6+1/20*(3*A*b-4 
*B*a)*(c*x^2+b*x+a)^(3/2)/a^2/x^5-1/160*(-20*A*a*c+21*A*b^2-28*B*a*b)*(c*x 
^2+b*x+a)^(3/2)/a^3/x^4+1/960*(-196*A*a*b*c+105*A*b^3+128*B*a^2*c-140*B*a* 
b^2)*(c*x^2+b*x+a)^(3/2)/a^4/x^3-1/1024*(-4*a*c+b^2)*(4*a*b*B*(-12*a*c+7*b 
^2)-A*(16*a^2*c^2-56*a*b^2*c+21*b^4))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2 
+b*x+a)^(1/2))/a^(11/2)
 

Mathematica [A] (verified)

Time = 3.55 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx=\frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (315 A b^5 x^5+256 a^5 (5 A+6 B x)-210 a b^3 x^4 (2 b B x+A (b+8 c x))+64 a^4 x (A (2 b+5 c x)+B x (3 b+8 c x))-16 a^3 x^2 \left (A \left (9 b^2+34 b c x+30 c^2 x^2\right )+2 B x \left (7 b^2+29 b c x+32 c^2 x^2\right )\right )+8 a^2 b x^3 \left (5 b B x (7 b+46 c x)+A \left (21 b^2+112 b c x+226 c^2 x^2\right )\right )\right )-315 A b^6 x^6 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-60 a \left (7 b^5 B+35 A b^4 c-40 a b^3 B c-60 a A b^2 c^2+48 a^2 b B c^2+16 a^2 A c^3\right ) x^6 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{7680 a^{11/2} x^6} \] Input:

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

Output:

(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(315*A*b^5*x^5 + 256*a^5*(5*A + 6*B*x) - 
210*a*b^3*x^4*(2*b*B*x + A*(b + 8*c*x)) + 64*a^4*x*(A*(2*b + 5*c*x) + B*x* 
(3*b + 8*c*x)) - 16*a^3*x^2*(A*(9*b^2 + 34*b*c*x + 30*c^2*x^2) + 2*B*x*(7* 
b^2 + 29*b*c*x + 32*c^2*x^2)) + 8*a^2*b*x^3*(5*b*B*x*(7*b + 46*c*x) + A*(2 
1*b^2 + 112*b*c*x + 226*c^2*x^2)))) - 315*A*b^6*x^6*ArcTanh[(Sqrt[c]*x - S 
qrt[a + x*(b + c*x)])/Sqrt[a]] - 60*a*(7*b^5*B + 35*A*b^4*c - 40*a*b^3*B*c 
 - 60*a*A*b^2*c^2 + 48*a^2*b*B*c^2 + 16*a^2*A*c^3)*x^6*ArcTanh[(-(Sqrt[c]* 
x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(7680*a^(11/2)*x^6)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1237, 27, 1237, 27, 1237, 27, 1228, 1152, 1154, 219}

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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {\left (21 A b^2-28 a B b-20 a A c+4 (3 A b-4 a B) c x\right ) \sqrt {c x^2+b x+a}}{2 x^5}dx}{5 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\left (21 A b^2-28 a B b-20 a A c+4 (3 A b-4 a B) c x\right ) \sqrt {c x^2+b x+a}}{x^5}dx}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (105 A b^3-140 a B b^2-196 a A c b+128 a^2 B c+2 c \left (21 A b^2-28 a B b-20 a A c\right ) x\right ) \sqrt {c x^2+b x+a}}{2 x^4}dx}{4 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{4 a x^4}}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (105 A b^3-140 a B b^2-196 a A c b+128 a^2 B c+2 c \left (21 A b^2-28 a B b-20 a A c\right ) x\right ) \sqrt {c x^2+b x+a}}{x^4}dx}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{4 a x^4}}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {\frac {5 \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{3 a x^3}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{4 a x^4}}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {-\frac {\frac {5 \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{3 a x^3}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{4 a x^4}}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {-\frac {\frac {5 \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{3 a x^3}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{4 a x^4}}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {-\frac {\frac {5 \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{3 a x^3}}{8 a}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{4 a x^4}}{10 a}-\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{5 a x^5}}{4 a}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}\)

Input:

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

Output:

-1/6*(A*(a + b*x + c*x^2)^(3/2))/(a*x^6) - (-1/5*((3*A*b - 4*a*B)*(a + b*x 
 + c*x^2)^(3/2))/(a*x^5) - (-1/4*((21*A*b^2 - 28*a*b*B - 20*a*A*c)*(a + b* 
x + c*x^2)^(3/2))/(a*x^4) - (-1/3*((105*A*b^3 - 140*a*b^2*B - 196*a*A*b*c 
+ 128*a^2*B*c)*(a + b*x + c*x^2)^(3/2))/(a*x^3) + (5*(4*a*b*B*(7*b^2 - 12* 
a*c) - A*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2))*(-1/4*((2*a + b*x)*Sqrt[a + b 
*x + c*x^2])/(a*x^2) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[ 
a + b*x + c*x^2])])/(8*a^(3/2))))/(2*a))/(8*a))/(10*a))/(4*a)
 

Defintions of rubi rules used

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 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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.11

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (1808 A \,a^{2} b \,c^{2} x^{5}-1680 A a \,b^{3} c \,x^{5}+315 A \,b^{5} x^{5}-1024 B \,a^{3} c^{2} x^{5}+1840 B \,a^{2} b^{2} c \,x^{5}-420 B a \,b^{4} x^{5}-480 A \,a^{3} c^{2} x^{4}+896 A \,a^{2} b^{2} c \,x^{4}-210 A a \,b^{4} x^{4}-928 B \,a^{3} b c \,x^{4}+280 B \,a^{2} b^{3} x^{4}-544 A \,a^{3} b c \,x^{3}+168 A \,a^{2} b^{3} x^{3}+512 B \,a^{4} c \,x^{3}-224 B \,a^{3} b^{2} x^{3}+320 A \,a^{4} c \,x^{2}-144 A \,a^{3} b^{2} x^{2}+192 B \,a^{4} b \,x^{2}+128 A \,a^{4} b x +1536 B \,a^{5} x +1280 A \,a^{5}\right )}{7680 x^{6} a^{5}}-\frac {\left (64 a^{3} A \,c^{3}-240 A \,a^{2} b^{2} c^{2}+140 A a \,b^{4} c -21 A \,b^{6}+192 B \,a^{3} b \,c^{2}-160 B \,a^{2} b^{3} c +28 B a \,b^{5}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{1024 a^{\frac {11}{2}}}\)	\(344\)
default	\(\text {Expression too large to display}\)	\(2627\)

Input:

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

Output:

-1/7680*(c*x^2+b*x+a)^(1/2)*(1808*A*a^2*b*c^2*x^5-1680*A*a*b^3*c*x^5+315*A 
*b^5*x^5-1024*B*a^3*c^2*x^5+1840*B*a^2*b^2*c*x^5-420*B*a*b^4*x^5-480*A*a^3 
*c^2*x^4+896*A*a^2*b^2*c*x^4-210*A*a*b^4*x^4-928*B*a^3*b*c*x^4+280*B*a^2*b 
^3*x^4-544*A*a^3*b*c*x^3+168*A*a^2*b^3*x^3+512*B*a^4*c*x^3-224*B*a^3*b^2*x 
^3+320*A*a^4*c*x^2-144*A*a^3*b^2*x^2+192*B*a^4*b*x^2+128*A*a^4*b*x+1536*B* 
a^5*x+1280*A*a^5)/x^6/a^5-1/1024*(64*A*a^3*c^3-240*A*a^2*b^2*c^2+140*A*a*b 
^4*c-21*A*b^6+192*B*a^3*b*c^2-160*B*a^2*b^3*c+28*B*a*b^5)/a^(11/2)*ln((2*a 
+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.29 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/30720*(15*(28*B*a*b^5 - 21*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 5*A*a 
^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 7*A*a*b^4)*c)*sqrt(a)*x^6*log(-(8*a*b*x + 
(b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x 
^2) - 4*(1280*A*a^6 - (420*B*a^2*b^4 - 315*A*a*b^5 + 16*(64*B*a^4 - 113*A* 
a^3*b)*c^2 - 80*(23*B*a^3*b^2 - 21*A*a^2*b^3)*c)*x^5 + 2*(140*B*a^3*b^3 - 
105*A*a^2*b^4 - 240*A*a^4*c^2 - 16*(29*B*a^4*b - 28*A*a^3*b^2)*c)*x^4 - 8* 
(28*B*a^4*b^2 - 21*A*a^3*b^3 - 4*(16*B*a^5 - 17*A*a^4*b)*c)*x^3 + 16*(12*B 
*a^5*b - 9*A*a^4*b^2 + 20*A*a^5*c)*x^2 + 128*(12*B*a^6 + A*a^5*b)*x)*sqrt( 
c*x^2 + b*x + a))/(a^6*x^6), 1/15360*(15*(28*B*a*b^5 - 21*A*b^6 + 64*A*a^3 
*c^3 + 48*(4*B*a^3*b - 5*A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 7*A*a*b^4)*c)* 
sqrt(-a)*x^6*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^ 
2 + a*b*x + a^2)) - 2*(1280*A*a^6 - (420*B*a^2*b^4 - 315*A*a*b^5 + 16*(64* 
B*a^4 - 113*A*a^3*b)*c^2 - 80*(23*B*a^3*b^2 - 21*A*a^2*b^3)*c)*x^5 + 2*(14 
0*B*a^3*b^3 - 105*A*a^2*b^4 - 240*A*a^4*c^2 - 16*(29*B*a^4*b - 28*A*a^3*b^ 
2)*c)*x^4 - 8*(28*B*a^4*b^2 - 21*A*a^3*b^3 - 4*(16*B*a^5 - 17*A*a^4*b)*c)* 
x^3 + 16*(12*B*a^5*b - 9*A*a^4*b^2 + 20*A*a^5*c)*x^2 + 128*(12*B*a^6 + A*a 
^5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^6)]
 

Sympy [F]

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

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

Output:

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1955 vs. \(2 (280) = 560\).

Time = 0.25 (sec) , antiderivative size = 1955, normalized size of antiderivative = 6.31 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx=\text {Too large to display} \] Input:

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

Output:

1/512*(28*B*a*b^5 - 21*A*b^6 - 160*B*a^2*b^3*c + 140*A*a*b^4*c + 192*B*a^3 
*b*c^2 - 240*A*a^2*b^2*c^2 + 64*A*a^3*c^3)*arctan(-(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^5) - 1/7680*(420*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^11*B*a*b^5 - 315*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*b^ 
6 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^2*b^3*c + 2100*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^11*A*a*b^4*c + 2880*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^11*B*a^3*b*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A* 
a^2*b^2*c^2 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^3*c^3 - 2380* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^2*b^5 + 1785*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^9*A*a*b^6 + 13600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B 
*a^3*b^3*c - 11900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b^4*c - 163 
20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^4*b*c^2 + 20400*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^9*A*a^3*b^2*c^2 - 5440*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^9*A*a^4*c^3 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^5*c^ 
(5/2) + 5544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^5 - 4158*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^6 - 31680*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^7*B*a^4*b^3*c + 27720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A* 
a^3*b^4*c - 48000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^5*b*c^2 - 4752 
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^4*b^2*c^2 - 36480*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^7*A*a^5*c^3 - 97280*(sqrt(c)*x - sqrt(c*x^2 + b...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.51 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx=\frac {-2560 \sqrt {c \,x^{2}+b x +a}\, a^{6}-3328 \sqrt {c \,x^{2}+b x +a}\, a^{5} b x -640 \sqrt {c \,x^{2}+b x +a}\, a^{5} c \,x^{2}-96 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{2} x^{2}+64 \sqrt {c \,x^{2}+b x +a}\, a^{4} b c \,x^{3}+960 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{2} x^{4}+112 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} x^{3}+64 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c \,x^{4}-1568 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2} x^{5}-140 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} x^{4}-320 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c \,x^{5}+210 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} x^{5}+960 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{3} c^{3} x^{6}-720 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b^{2} c^{2} x^{6}-300 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{4} c \,x^{6}+105 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{6} x^{6}-960 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} c^{3} x^{6}+720 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b^{2} c^{2} x^{6}+300 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{4} c \,x^{6}-105 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{6} x^{6}}{15360 a^{5} x^{6}} \] Input:

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

Output:

( - 2560*sqrt(a + b*x + c*x**2)*a**6 - 3328*sqrt(a + b*x + c*x**2)*a**5*b* 
x - 640*sqrt(a + b*x + c*x**2)*a**5*c*x**2 - 96*sqrt(a + b*x + c*x**2)*a** 
4*b**2*x**2 + 64*sqrt(a + b*x + c*x**2)*a**4*b*c*x**3 + 960*sqrt(a + b*x + 
 c*x**2)*a**4*c**2*x**4 + 112*sqrt(a + b*x + c*x**2)*a**3*b**3*x**3 + 64*s 
qrt(a + b*x + c*x**2)*a**3*b**2*c*x**4 - 1568*sqrt(a + b*x + c*x**2)*a**3* 
b*c**2*x**5 - 140*sqrt(a + b*x + c*x**2)*a**2*b**4*x**4 - 320*sqrt(a + b*x 
 + c*x**2)*a**2*b**3*c*x**5 + 210*sqrt(a + b*x + c*x**2)*a*b**5*x**5 + 960 
*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**3*c**3*x**6 
- 720*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**2* 
c**2*x**6 - 300*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)* 
a*b**4*c*x**6 + 105*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b 
*x)*b**6*x**6 - 960*sqrt(a)*log(x)*a**3*c**3*x**6 + 720*sqrt(a)*log(x)*a** 
2*b**2*c**2*x**6 + 300*sqrt(a)*log(x)*a*b**4*c*x**6 - 105*sqrt(a)*log(x)*b 
**6*x**6)/(15360*a**5*x**6)