Integrand size = 19, antiderivative size = 230 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=-\frac {a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}+\frac {2 \left (a^2-b^2 c\right )^3 \sqrt {c+d x}}{b^7 d^4}+\frac {2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac {a (c+d x)^3}{3 b^2 d^4}+\frac {2 (c+d x)^{7/2}}{7 b d^4}-\frac {2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4} \]
-a*(3*b^4*c^2-3*a^2*b^2*c+a^4)*x/b^6/d^3+2/3*(3*b^4*c^2-3*a^2*b^2*c+a^4)*( d*x+c)^(3/2)/b^5/d^4-1/2*a*(-3*b^2*c+a^2)*(d*x+c)^2/b^4/d^4+2/5*(-3*b^2*c+ a^2)*(d*x+c)^(5/2)/b^3/d^4-1/3*a*(d*x+c)^3/b^2/d^4+2/7*(d*x+c)^(7/2)/b/d^4 -2*a*(-b^2*c+a^2)^3*ln(a+b*(d*x+c)^(1/2))/b^8/d^4+2*(-b^2*c+a^2)^3*(d*x+c) ^(1/2)/b^7/d^4
Time = 0.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=\frac {b \left (420 a^6 \sqrt {c+d x}-140 a^4 b^2 (8 c-d x) \sqrt {c+d x}-210 a^5 b (c+d x)+105 a^3 b^3 \left (5 c^2+4 c d x-d^2 x^2\right )+84 a^2 b^4 \sqrt {c+d x} \left (11 c^2-3 c d x+d^2 x^2\right )-35 a b^5 \left (11 c^3+6 c^2 d x-3 c d^2 x^2+2 d^3 x^3\right )+12 b^6 \sqrt {c+d x} \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )\right )-420 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{210 b^8 d^4} \]
(b*(420*a^6*Sqrt[c + d*x] - 140*a^4*b^2*(8*c - d*x)*Sqrt[c + d*x] - 210*a^ 5*b*(c + d*x) + 105*a^3*b^3*(5*c^2 + 4*c*d*x - d^2*x^2) + 84*a^2*b^4*Sqrt[ c + d*x]*(11*c^2 - 3*c*d*x + d^2*x^2) - 35*a*b^5*(11*c^3 + 6*c^2*d*x - 3*c *d^2*x^2 + 2*d^3*x^3) + 12*b^6*Sqrt[c + d*x]*(-16*c^3 + 8*c^2*d*x - 6*c*d^ 2*x^2 + 5*d^3*x^3)) - 420*a*(a^2 - b^2*c)^3*Log[a + b*Sqrt[c + d*x]])/(210 *b^8*d^4)
Time = 0.41 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {896, 25, 1732, 522, 2009}
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.
| \(\displaystyle \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int \frac {d^3 x^3}{a+b \sqrt {c+d x}}d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d^3 x^3}{a+b \sqrt {c+d x}}d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 1732 |
| \(\displaystyle -\frac {2 \int -\frac {d^3 x^3 \sqrt {c+d x}}{a+b \sqrt {c+d x}}d\sqrt {c+d x}}{d^4}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {2 \int \left (\frac {a \left (a^2-b^2 c\right )^3}{b^7 \left (a+b \sqrt {c+d x}\right )}+\frac {\left (b^2 c-a^2\right )^3}{b^7}-\frac {(c+d x)^3}{b}+\frac {a (c+d x)^{5/2}}{b^2}+\frac {\left (3 b^2 c-a^2\right ) (c+d x)^2}{b^3}+\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^{3/2}}{b^4}-\frac {\left (a^4-3 b^2 c a^2+3 b^4 c^2\right ) (c+d x)}{b^5}+\frac {a \left (a^4-3 b^2 c a^2+3 b^4 c^2\right ) \sqrt {c+d x}}{b^6}\right )d\sqrt {c+d x}}{d^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (\frac {a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{b^8}-\frac {\left (a^2-b^2 c\right )^3 \sqrt {c+d x}}{b^7}+\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^2}{4 b^4}-\frac {\left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3}+\frac {a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)}{2 b^6}-\frac {\left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5}+\frac {a (c+d x)^3}{6 b^2}-\frac {(c+d x)^{7/2}}{7 b}\right )}{d^4}\) |
(-2*(-(((a^2 - b^2*c)^3*Sqrt[c + d*x])/b^7) + (a*(a^4 - 3*a^2*b^2*c + 3*b^ 4*c^2)*(c + d*x))/(2*b^6) - ((a^4 - 3*a^2*b^2*c + 3*b^4*c^2)*(c + d*x)^(3/ 2))/(3*b^5) + (a*(a^2 - 3*b^2*c)*(c + d*x)^2)/(4*b^4) - ((a^2 - 3*b^2*c)*( c + d*x)^(5/2))/(5*b^3) + (a*(c + d*x)^3)/(6*b^2) - (c + d*x)^(7/2)/(7*b) + (a*(a^2 - b^2*c)^3*Log[a + b*Sqrt[c + d*x]])/b^8))/d^4
3.7.32.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(d + e*x^(g* n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} , x] && EqQ[n2, 2*n] && FractionQ[n]
Time = 0.06 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {7}{2}} b^{6}}{7}-\frac {a \left (d x +c \right )^{3} b^{5}}{6}-\frac {3 b^{6} c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {a^{2} b^{4} \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{5} c \left (d x +c \right )^{2}}{4}+b^{6} c^{2} \left (d x +c \right )^{\frac {3}{2}}-\frac {a^{3} b^{3} \left (d x +c \right )^{2}}{4}-a^{2} b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 a \,b^{5} c^{2} \left (d x +c \right )}{2}-b^{6} c^{3} \sqrt {d x +c}+\frac {a^{4} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {3 a^{3} b^{3} c \left (d x +c \right )}{2}+3 a^{2} b^{4} c^{2} \sqrt {d x +c}-\frac {a^{5} b \left (d x +c \right )}{2}-3 a^{4} b^{2} c \sqrt {d x +c}+a^{6} \sqrt {d x +c}\right )}{b^{7}}-\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}}{d^{4}}\) | \(288\) |
| default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {7}{2}} b^{6}}{7}-\frac {a \left (d x +c \right )^{3} b^{5}}{6}-\frac {3 b^{6} c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {a^{2} b^{4} \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{5} c \left (d x +c \right )^{2}}{4}+b^{6} c^{2} \left (d x +c \right )^{\frac {3}{2}}-\frac {a^{3} b^{3} \left (d x +c \right )^{2}}{4}-a^{2} b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 a \,b^{5} c^{2} \left (d x +c \right )}{2}-b^{6} c^{3} \sqrt {d x +c}+\frac {a^{4} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {3 a^{3} b^{3} c \left (d x +c \right )}{2}+3 a^{2} b^{4} c^{2} \sqrt {d x +c}-\frac {a^{5} b \left (d x +c \right )}{2}-3 a^{4} b^{2} c \sqrt {d x +c}+a^{6} \sqrt {d x +c}\right )}{b^{7}}-\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}}{d^{4}}\) | \(288\) |
2/d^4*(1/b^7*(1/7*(d*x+c)^(7/2)*b^6-1/6*a*(d*x+c)^3*b^5-3/5*b^6*c*(d*x+c)^ (5/2)+1/5*a^2*b^4*(d*x+c)^(5/2)+3/4*a*b^5*c*(d*x+c)^2+b^6*c^2*(d*x+c)^(3/2 )-1/4*a^3*b^3*(d*x+c)^2-a^2*b^4*c*(d*x+c)^(3/2)-3/2*a*b^5*c^2*(d*x+c)-b^6* c^3*(d*x+c)^(1/2)+1/3*a^4*b^2*(d*x+c)^(3/2)+3/2*a^3*b^3*c*(d*x+c)+3*a^2*b^ 4*c^2*(d*x+c)^(1/2)-1/2*a^5*b*(d*x+c)-3*a^4*b^2*c*(d*x+c)^(1/2)+a^6*(d*x+c )^(1/2))-a*(-b^6*c^3+3*a^2*b^4*c^2-3*a^4*b^2*c+a^6)/b^8*ln(a+b*(d*x+c)^(1/ 2)))
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.99 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=-\frac {70 \, a b^{6} d^{3} x^{3} - 105 \, {\left (a b^{6} c - a^{3} b^{4}\right )} d^{2} x^{2} + 210 \, {\left (a b^{6} c^{2} - 2 \, a^{3} b^{4} c + a^{5} b^{2}\right )} d x - 420 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (15 \, b^{7} d^{3} x^{3} - 48 \, b^{7} c^{3} + 231 \, a^{2} b^{5} c^{2} - 280 \, a^{4} b^{3} c + 105 \, a^{6} b - 3 \, {\left (6 \, b^{7} c - 7 \, a^{2} b^{5}\right )} d^{2} x^{2} + {\left (24 \, b^{7} c^{2} - 63 \, a^{2} b^{5} c + 35 \, a^{4} b^{3}\right )} d x\right )} \sqrt {d x + c}}{210 \, b^{8} d^{4}} \]
-1/210*(70*a*b^6*d^3*x^3 - 105*(a*b^6*c - a^3*b^4)*d^2*x^2 + 210*(a*b^6*c^ 2 - 2*a^3*b^4*c + a^5*b^2)*d*x - 420*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^ 2*c - a^7)*log(sqrt(d*x + c)*b + a) - 4*(15*b^7*d^3*x^3 - 48*b^7*c^3 + 231 *a^2*b^5*c^2 - 280*a^4*b^3*c + 105*a^6*b - 3*(6*b^7*c - 7*a^2*b^5)*d^2*x^2 + (24*b^7*c^2 - 63*a^2*b^5*c + 35*a^4*b^3)*d*x)*sqrt(d*x + c))/(b^8*d^4)
Time = 1.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (- \frac {a \left (c + d x\right )^{3}}{6 b^{2}} - \frac {a \left (a^{2} - b^{2} c\right )^{3} \left (\begin {cases} \frac {\sqrt {c + d x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {c + d x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{7}} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{\frac {5}{2}}}{5 b^{3}} + \frac {\left (- a^{3} + 3 a b^{2} c\right ) \left (c + d x\right )^{2}}{4 b^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{4} - 3 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{3 b^{5}} + \frac {\left (c + d x\right ) \left (- a^{5} + 3 a^{3} b^{2} c - 3 a b^{4} c^{2}\right )}{2 b^{6}} + \frac {\sqrt {c + d x} \left (a^{6} - 3 a^{4} b^{2} c + 3 a^{2} b^{4} c^{2} - b^{6} c^{3}\right )}{b^{7}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4}}{4 \left (a + b \sqrt {c}\right )} & \text {otherwise} \end {cases} \]
Piecewise((2*(-a*(c + d*x)**3/(6*b**2) - a*(a**2 - b**2*c)**3*Piecewise((s qrt(c + d*x)/a, Eq(b, 0)), (log(a + b*sqrt(c + d*x))/b, True))/b**7 + (c + d*x)**(7/2)/(7*b) + (a**2 - 3*b**2*c)*(c + d*x)**(5/2)/(5*b**3) + (-a**3 + 3*a*b**2*c)*(c + d*x)**2/(4*b**4) + (c + d*x)**(3/2)*(a**4 - 3*a**2*b**2 *c + 3*b**4*c**2)/(3*b**5) + (c + d*x)*(-a**5 + 3*a**3*b**2*c - 3*a*b**4*c **2)/(2*b**6) + sqrt(c + d*x)*(a**6 - 3*a**4*b**2*c + 3*a**2*b**4*c**2 - b **6*c**3)/b**7)/d**4, Ne(d, 0)), (x**4/(4*(a + b*sqrt(c))), True))
Time = 0.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=\frac {\frac {60 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} - 70 \, {\left (d x + c\right )}^{3} a b^{5} - 84 \, {\left (3 \, b^{6} c - a^{2} b^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left (3 \, a b^{5} c - a^{3} b^{3}\right )} {\left (d x + c\right )}^{2} + 140 \, {\left (3 \, b^{6} c^{2} - 3 \, a^{2} b^{4} c + a^{4} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 210 \, {\left (3 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c + a^{5} b\right )} {\left (d x + c\right )} - 420 \, {\left (b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}\right )} \sqrt {d x + c}}{b^{7}} + \frac {420 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8}}}{210 \, d^{4}} \]
1/210*((60*(d*x + c)^(7/2)*b^6 - 70*(d*x + c)^3*a*b^5 - 84*(3*b^6*c - a^2* b^4)*(d*x + c)^(5/2) + 105*(3*a*b^5*c - a^3*b^3)*(d*x + c)^2 + 140*(3*b^6* c^2 - 3*a^2*b^4*c + a^4*b^2)*(d*x + c)^(3/2) - 210*(3*a*b^5*c^2 - 3*a^3*b^ 3*c + a^5*b)*(d*x + c) - 420*(b^6*c^3 - 3*a^2*b^4*c^2 + 3*a^4*b^2*c - a^6) *sqrt(d*x + c))/b^7 + 420*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)* log(sqrt(d*x + c)*b + a)/b^8)/d^4
Time = 0.42 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.48 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=\frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} + \frac {60 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} d^{24} - 252 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{6} c d^{24} + 420 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{6} c^{2} d^{24} - 420 \, \sqrt {d x + c} b^{6} c^{3} d^{24} - 70 \, {\left (d x + c\right )}^{3} a b^{5} d^{24} + 315 \, {\left (d x + c\right )}^{2} a b^{5} c d^{24} - 630 \, {\left (d x + c\right )} a b^{5} c^{2} d^{24} + 84 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{4} d^{24} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{4} c d^{24} + 1260 \, \sqrt {d x + c} a^{2} b^{4} c^{2} d^{24} - 105 \, {\left (d x + c\right )}^{2} a^{3} b^{3} d^{24} + 630 \, {\left (d x + c\right )} a^{3} b^{3} c d^{24} + 140 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{2} d^{24} - 1260 \, \sqrt {d x + c} a^{4} b^{2} c d^{24} - 210 \, {\left (d x + c\right )} a^{5} b d^{24} + 420 \, \sqrt {d x + c} a^{6} d^{24}}{210 \, b^{7} d^{28}} \]
2*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*log(abs(sqrt(d*x + c)*b + a))/(b^8*d^4) + 1/210*(60*(d*x + c)^(7/2)*b^6*d^24 - 252*(d*x + c)^(5/2) *b^6*c*d^24 + 420*(d*x + c)^(3/2)*b^6*c^2*d^24 - 420*sqrt(d*x + c)*b^6*c^3 *d^24 - 70*(d*x + c)^3*a*b^5*d^24 + 315*(d*x + c)^2*a*b^5*c*d^24 - 630*(d* x + c)*a*b^5*c^2*d^24 + 84*(d*x + c)^(5/2)*a^2*b^4*d^24 - 420*(d*x + c)^(3 /2)*a^2*b^4*c*d^24 + 1260*sqrt(d*x + c)*a^2*b^4*c^2*d^24 - 105*(d*x + c)^2 *a^3*b^3*d^24 + 630*(d*x + c)*a^3*b^3*c*d^24 + 140*(d*x + c)^(3/2)*a^4*b^2 *d^24 - 1260*sqrt(d*x + c)*a^4*b^2*c*d^24 - 210*(d*x + c)*a^5*b*d^24 + 420 *sqrt(d*x + c)*a^6*d^24)/(b^7*d^28)
Time = 0.07 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.38 \[ \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx=\frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,b\,d^4}-\left (\frac {a^2\,\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{b^2}-\frac {6\,c^2}{b\,d^4}\right )}{b^2}+\frac {2\,c^3}{b\,d^4}\right )\,\sqrt {c+d\,x}-\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{3\,b^2}-\frac {2\,c^2}{b\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {6\,c}{5\,b\,d^4}-\frac {2\,a^2}{5\,b^3\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {a\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )\,{\left (c+d\,x\right )}^2}{4\,b}-\frac {a\,{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}-\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (2\,a^7-6\,a^5\,b^2\,c+6\,a^3\,b^4\,c^2-2\,a\,b^6\,c^3\right )}{b^8\,d^4}+\frac {a\,d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{b^2}-\frac {6\,c^2}{b\,d^4}\right )}{2\,b} \]
(2*(c + d*x)^(7/2))/(7*b*d^4) - ((a^2*((a^2*((6*c)/(b*d^4) - (2*a^2)/(b^3* d^4)))/b^2 - (6*c^2)/(b*d^4)))/b^2 + (2*c^3)/(b*d^4))*(c + d*x)^(1/2) - (( a^2*((6*c)/(b*d^4) - (2*a^2)/(b^3*d^4)))/(3*b^2) - (2*c^2)/(b*d^4))*(c + d *x)^(3/2) - ((6*c)/(5*b*d^4) - (2*a^2)/(5*b^3*d^4))*(c + d*x)^(5/2) + (a*( (6*c)/(b*d^4) - (2*a^2)/(b^3*d^4))*(c + d*x)^2)/(4*b) - (a*(c + d*x)^3)/(3 *b^2*d^4) - (log(a + b*(c + d*x)^(1/2))*(2*a^7 - 6*a^5*b^2*c - 2*a*b^6*c^3 + 6*a^3*b^4*c^2))/(b^8*d^4) + (a*d*x*((a^2*((6*c)/(b*d^4) - (2*a^2)/(b^3* d^4)))/b^2 - (6*c^2)/(b*d^4)))/(2*b)