\(\int x^2 (a x+b x^2)^{5/2} \, dx\) [22]

Optimal result

Integrand size = 17, antiderivative size = 227 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {45 a^7 \sqrt {a x+b x^2}}{16384 b^5}-\frac {15 a^6 x \sqrt {a x+b x^2}}{8192 b^4}+\frac {3 a^5 x^2 \sqrt {a x+b x^2}}{2048 b^3}-\frac {9 a^4 x^3 \sqrt {a x+b x^2}}{7168 b^2}+\frac {a^3 x^4 \sqrt {a x+b x^2}}{896 b}+\frac {81}{448} a^2 x^5 \sqrt {a x+b x^2}+\frac {33}{112} a b x^6 \sqrt {a x+b x^2}+\frac {1}{8} b^2 x^7 \sqrt {a x+b x^2}-\frac {45 a^8 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{16384 b^{11/2}} \] Output:

45/16384*a^7*(b*x^2+a*x)^(1/2)/b^5-15/8192*a^6*x*(b*x^2+a*x)^(1/2)/b^4+3/2 
048*a^5*x^2*(b*x^2+a*x)^(1/2)/b^3-9/7168*a^4*x^3*(b*x^2+a*x)^(1/2)/b^2+1/8 
96*a^3*x^4*(b*x^2+a*x)^(1/2)/b+81/448*a^2*x^5*(b*x^2+a*x)^(1/2)+33/112*a*b 
*x^6*(b*x^2+a*x)^(1/2)+1/8*b^2*x^7*(b*x^2+a*x)^(1/2)-45/16384*a^8*arctanh( 
b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.67 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {\sqrt {x (a+b x)} \left (\sqrt {b} \left (315 a^7-210 a^6 b x+168 a^5 b^2 x^2-144 a^4 b^3 x^3+128 a^3 b^4 x^4+20736 a^2 b^5 x^5+33792 a b^6 x^6+14336 b^7 x^7\right )+\frac {630 a^8 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{\sqrt {x} \sqrt {a+b x}}\right )}{114688 b^{11/2}} \] Input:

Integrate[x^2*(a*x + b*x^2)^(5/2),x]
 

Output:

(Sqrt[x*(a + b*x)]*(Sqrt[b]*(315*a^7 - 210*a^6*b*x + 168*a^5*b^2*x^2 - 144 
*a^4*b^3*x^3 + 128*a^3*b^4*x^4 + 20736*a^2*b^5*x^5 + 33792*a*b^6*x^6 + 143 
36*b^7*x^7) + (630*a^8*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x]) 
])/(Sqrt[x]*Sqrt[a + b*x])))/(114688*b^(11/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.85, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {1134, 1160, 1087, 1087, 1087, 1091, 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.

\(\displaystyle \int x^2 \left (a x+b x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \int x \left (b x^2+a x\right )^{5/2}dx}{16 b}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \left (\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \int \left (b x^2+a x\right )^{5/2}dx}{2 b}\right )}{16 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \left (\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^2 \int \left (b x^2+a x\right )^{3/2}dx}{24 b}\right )}{2 b}\right )}{16 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \left (\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^2 \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \int \sqrt {b x^2+a x}dx}{16 b}\right )}{24 b}\right )}{2 b}\right )}{16 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \left (\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^2 \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b}\right )}{24 b}\right )}{2 b}\right )}{16 b}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \left (\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^2 \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b}\right )}{24 b}\right )}{2 b}\right )}{16 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x \left (a x+b x^2\right )^{7/2}}{8 b}-\frac {9 a \left (\frac {\left (a x+b x^2\right )^{7/2}}{7 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac {5 a^2 \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right )}{16 b}\right )}{24 b}\right )}{2 b}\right )}{16 b}\)

Input:

Int[x^2*(a*x + b*x^2)^(5/2),x]
 

Output:

(x*(a*x + b*x^2)^(7/2))/(8*b) - (9*a*((a*x + b*x^2)^(7/2)/(7*b) - (a*(((a 
+ 2*b*x)*(a*x + b*x^2)^(5/2))/(12*b) - (5*a^2*(((a + 2*b*x)*(a*x + b*x^2)^ 
(3/2))/(8*b) - (3*a^2*(((a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b) - (a^2*ArcTan 
h[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*b^(3/2))))/(16*b)))/(24*b)))/(2*b)))/ 
(16*b)
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1091

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

rule 1134

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

rule 1160

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {\left (14336 x^{7} b^{7}+33792 b^{6} a \,x^{6}+20736 a^{2} x^{5} b^{5}+128 b^{4} x^{4} a^{3}-144 b^{3} x^{3} a^{4}+168 x^{2} b^{2} a^{5}-210 x b \,a^{6}+315 a^{7}\right ) x \left (b x +a \right )}{114688 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {45 a^{8} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{32768 b^{\frac {11}{2}}}\)	\(128\)
default	\(\frac {x \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{8 b}-\frac {9 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{7 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{12 b}-\frac {5 a^{2} \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{24 b}\right )}{2 b}\right )}{16 b}\)	\(165\)

Input:

int(x^2*(b*x^2+a*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/114688*(14336*b^7*x^7+33792*a*b^6*x^6+20736*a^2*b^5*x^5+128*a^3*b^4*x^4- 
144*a^4*b^3*x^3+168*a^5*b^2*x^2-210*a^6*b*x+315*a^7)*x*(b*x+a)/b^5/(x*(b*x 
+a))^(1/2)-45/32768*a^8/b^(11/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.14 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\left [\frac {315 \, a^{8} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (14336 \, b^{8} x^{7} + 33792 \, a b^{7} x^{6} + 20736 \, a^{2} b^{6} x^{5} + 128 \, a^{3} b^{5} x^{4} - 144 \, a^{4} b^{4} x^{3} + 168 \, a^{5} b^{3} x^{2} - 210 \, a^{6} b^{2} x + 315 \, a^{7} b\right )} \sqrt {b x^{2} + a x}}{229376 \, b^{6}}, \frac {315 \, a^{8} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (14336 \, b^{8} x^{7} + 33792 \, a b^{7} x^{6} + 20736 \, a^{2} b^{6} x^{5} + 128 \, a^{3} b^{5} x^{4} - 144 \, a^{4} b^{4} x^{3} + 168 \, a^{5} b^{3} x^{2} - 210 \, a^{6} b^{2} x + 315 \, a^{7} b\right )} \sqrt {b x^{2} + a x}}{114688 \, b^{6}}\right ] \] Input:

integrate(x^2*(b*x^2+a*x)^(5/2),x, algorithm="fricas")
 

Output:

[1/229376*(315*a^8*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 
2*(14336*b^8*x^7 + 33792*a*b^7*x^6 + 20736*a^2*b^6*x^5 + 128*a^3*b^5*x^4 - 
 144*a^4*b^4*x^3 + 168*a^5*b^3*x^2 - 210*a^6*b^2*x + 315*a^7*b)*sqrt(b*x^2 
 + a*x))/b^6, 1/114688*(315*a^8*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b) 
/(b*x + a)) + (14336*b^8*x^7 + 33792*a*b^7*x^6 + 20736*a^2*b^6*x^5 + 128*a 
^3*b^5*x^4 - 144*a^4*b^4*x^3 + 168*a^5*b^3*x^2 - 210*a^6*b^2*x + 315*a^7*b 
)*sqrt(b*x^2 + a*x))/b^6]
 

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.82 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\begin {cases} - \frac {45 a^{8} \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{32768 b^{5}} + \sqrt {a x + b x^{2}} \cdot \left (\frac {45 a^{7}}{16384 b^{5}} - \frac {15 a^{6} x}{8192 b^{4}} + \frac {3 a^{5} x^{2}}{2048 b^{3}} - \frac {9 a^{4} x^{3}}{7168 b^{2}} + \frac {a^{3} x^{4}}{896 b} + \frac {81 a^{2} x^{5}}{448} + \frac {33 a b x^{6}}{112} + \frac {b^{2} x^{7}}{8}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (a x\right )^{\frac {11}{2}}}{11 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x**2*(b*x**2+a*x)**(5/2),x)
 

Output:

Piecewise((-45*a**8*Piecewise((log(a + 2*sqrt(b)*sqrt(a*x + b*x**2) + 2*b* 
x)/sqrt(b), Ne(a**2/b, 0)), ((a/(2*b) + x)*log(a/(2*b) + x)/sqrt(b*(a/(2*b 
) + x)**2), True))/(32768*b**5) + sqrt(a*x + b*x**2)*(45*a**7/(16384*b**5) 
 - 15*a**6*x/(8192*b**4) + 3*a**5*x**2/(2048*b**3) - 9*a**4*x**3/(7168*b** 
2) + a**3*x**4/(896*b) + 81*a**2*x**5/448 + 33*a*b*x**6/112 + b**2*x**7/8) 
, Ne(b, 0)), (2*(a*x)**(11/2)/(11*a**3), Ne(a, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {45 \, \sqrt {b x^{2} + a x} a^{6} x}{8192 \, b^{4}} - \frac {15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4} x}{1024 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{2} x}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {7}{2}} x}{8 \, b} - \frac {45 \, a^{8} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{32768 \, b^{\frac {11}{2}}} + \frac {45 \, \sqrt {b x^{2} + a x} a^{7}}{16384 \, b^{5}} - \frac {15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{5}}{2048 \, b^{4}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{3}}{128 \, b^{3}} - \frac {9 \, {\left (b x^{2} + a x\right )}^{\frac {7}{2}} a}{112 \, b^{2}} \] Input:

integrate(x^2*(b*x^2+a*x)^(5/2),x, algorithm="maxima")
 

Output:

45/8192*sqrt(b*x^2 + a*x)*a^6*x/b^4 - 15/1024*(b*x^2 + a*x)^(3/2)*a^4*x/b^ 
3 + 3/64*(b*x^2 + a*x)^(5/2)*a^2*x/b^2 + 1/8*(b*x^2 + a*x)^(7/2)*x/b - 45/ 
32768*a^8*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 45/16384 
*sqrt(b*x^2 + a*x)*a^7/b^5 - 15/2048*(b*x^2 + a*x)^(3/2)*a^5/b^4 + 3/128*( 
b*x^2 + a*x)^(5/2)*a^3/b^3 - 9/112*(b*x^2 + a*x)^(7/2)*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.57 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {45 \, a^{8} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{32768 \, b^{\frac {11}{2}}} + \frac {1}{114688} \, \sqrt {b x^{2} + a x} {\left (\frac {315 \, a^{7}}{b^{5}} - 2 \, {\left (\frac {105 \, a^{6}}{b^{4}} - 4 \, {\left (\frac {21 \, a^{5}}{b^{3}} - 2 \, {\left (\frac {9 \, a^{4}}{b^{2}} - 8 \, {\left (\frac {a^{3}}{b} + 2 \, {\left (81 \, a^{2} + 4 \, {\left (14 \, b^{2} x + 33 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

integrate(x^2*(b*x^2+a*x)^(5/2),x, algorithm="giac")
 

Output:

45/32768*a^8*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(11 
/2) + 1/114688*sqrt(b*x^2 + a*x)*(315*a^7/b^5 - 2*(105*a^6/b^4 - 4*(21*a^5 
/b^3 - 2*(9*a^4/b^2 - 8*(a^3/b + 2*(81*a^2 + 4*(14*b^2*x + 33*a*b)*x)*x)*x 
)*x)*x)*x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\int x^2\,{\left (b\,x^2+a\,x\right )}^{5/2} \,d x \] Input:

int(x^2*(a*x + b*x^2)^(5/2),x)
 

Output:

int(x^2*(a*x + b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.75 \[ \int x^2 \left (a x+b x^2\right )^{5/2} \, dx=\frac {315 \sqrt {x}\, \sqrt {b x +a}\, a^{7} b -210 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b^{2} x +168 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{3} x^{2}-144 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{4} x^{3}+128 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{5} x^{4}+20736 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{6} x^{5}+33792 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{7} x^{6}+14336 \sqrt {x}\, \sqrt {b x +a}\, b^{8} x^{7}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{8}}{114688 b^{6}} \] Input:

int(x^2*(b*x^2+a*x)^(5/2),x)
 

Output:

(315*sqrt(x)*sqrt(a + b*x)*a**7*b - 210*sqrt(x)*sqrt(a + b*x)*a**6*b**2*x 
+ 168*sqrt(x)*sqrt(a + b*x)*a**5*b**3*x**2 - 144*sqrt(x)*sqrt(a + b*x)*a** 
4*b**4*x**3 + 128*sqrt(x)*sqrt(a + b*x)*a**3*b**5*x**4 + 20736*sqrt(x)*sqr 
t(a + b*x)*a**2*b**6*x**5 + 33792*sqrt(x)*sqrt(a + b*x)*a*b**7*x**6 + 1433 
6*sqrt(x)*sqrt(a + b*x)*b**8*x**7 - 315*sqrt(b)*log((sqrt(a + b*x) + sqrt( 
x)*sqrt(b))/sqrt(a))*a**8)/(114688*b**6)