Integrand size = 19, antiderivative size = 240 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4}+\frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4} \] Output:
(3*b^4*c^2-9*a^2*b^2*c+5*a^4)*x/b^6/d^3-12*a*(-b^2*c+a^2)^2*(d*x+c)^(1/2)/ b^7/d^4-4/3*a*(-3*b^2*c+2*a^2)*(d*x+c)^(3/2)/b^5/d^4+3/2*(-b^2*c+a^2)*(d*x +c)^2/b^4/d^4-4/5*a*(d*x+c)^(5/2)/b^3/d^4+1/3*(d*x+c)^3/b^2/d^4+2*a*(-b^2* c+a^2)^3/b^8/d^4/(a+b*(d*x+c)^(1/2))+2*(-b^2*c+a^2)^2*(-b^2*c+7*a^2)*ln(a+ b*(d*x+c)^(1/2))/b^8/d^4
Time = 0.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {60 a^7-360 a^6 b \sqrt {c+d x}-30 a^5 b^2 (13 c+7 d x)+10 a^4 b^3 \sqrt {c+d x} (79 c+7 d x)-3 a^2 b^5 \sqrt {c+d x} \left (163 c^2+36 c d x-7 d^2 x^2\right )+5 a^3 b^4 \left (119 c^2+76 c d x-7 d^2 x^2\right )+5 b^7 \sqrt {c+d x} \left (11 c^3+6 c^2 d x-3 c d^2 x^2+2 d^3 x^3\right )-a b^6 \left (269 c^3+162 c^2 d x-33 c d^2 x^2+14 d^3 x^3\right )+60 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right ) \log \left (a+b \sqrt {c+d x}\right )}{30 b^8 d^4 \left (a+b \sqrt {c+d x}\right )} \] Input:
Integrate[x^3/(a + b*Sqrt[c + d*x])^2,x]
Output:
(60*a^7 - 360*a^6*b*Sqrt[c + d*x] - 30*a^5*b^2*(13*c + 7*d*x) + 10*a^4*b^3 *Sqrt[c + d*x]*(79*c + 7*d*x) - 3*a^2*b^5*Sqrt[c + d*x]*(163*c^2 + 36*c*d* x - 7*d^2*x^2) + 5*a^3*b^4*(119*c^2 + 76*c*d*x - 7*d^2*x^2) + 5*b^7*Sqrt[c + d*x]*(11*c^3 + 6*c^2*d*x - 3*c*d^2*x^2 + 2*d^3*x^3) - a*b^6*(269*c^3 + 162*c^2*d*x - 33*c*d^2*x^2 + 14*d^3*x^3) + 60*(a^2 - b^2*c)^2*(7*a^2 - b^2 *c)*(a + b*Sqrt[c + d*x])*Log[a + b*Sqrt[c + d*x]])/(30*b^8*d^4*(a + b*Sqr t[c + d*x]))
Time = 0.74 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int \frac {d^3 x^3}{\left (a+b \sqrt {c+d x}\right )^2}d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d^3 x^3}{\left (a+b \sqrt {c+d x}\right )^2}d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 1732 |
| \(\displaystyle -\frac {2 \int -\frac {d^3 x^3 \sqrt {c+d x}}{\left (a+b \sqrt {c+d x}\right )^2}d\sqrt {c+d x}}{d^4}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {2 \int \left (\frac {a \left (a^2-b^2 c\right )^3}{b^7 \left (a+b \sqrt {c+d x}\right )^2}+\frac {6 a \left (a^2-b^2 c\right )^2}{b^7}-\frac {(c+d x)^{5/2}}{b^2}+\frac {2 a (c+d x)^2}{b^3}+\frac {3 \left (b^2 c-a^2\right ) (c+d x)^{3/2}}{b^4}+\frac {2 a \left (2 a^2-3 b^2 c\right ) (c+d x)}{b^5}-\frac {\left (5 a^4-9 b^2 c a^2+3 b^4 c^2\right ) \sqrt {c+d x}}{b^6}+\frac {\left (b^2 c-7 a^2\right ) \left (b^2 c-a^2\right )^2}{b^7 \left (a+b \sqrt {c+d x}\right )}\right )d\sqrt {c+d x}}{d^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-\frac {a \left (a^2-b^2 c\right )^3}{b^8 \left (a+b \sqrt {c+d x}\right )}-\frac {\left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^8}+\frac {6 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7}+\frac {2 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5}-\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{4 b^4}-\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)}{2 b^6}+\frac {2 a (c+d x)^{5/2}}{5 b^3}-\frac {(c+d x)^3}{6 b^2}\right )}{d^4}\) |
Input:
Int[x^3/(a + b*Sqrt[c + d*x])^2,x]
Output:
(-2*((6*a*(a^2 - b^2*c)^2*Sqrt[c + d*x])/b^7 - ((5*a^4 - 9*a^2*b^2*c + 3*b ^4*c^2)*(c + d*x))/(2*b^6) + (2*a*(2*a^2 - 3*b^2*c)*(c + d*x)^(3/2))/(3*b^ 5) - (3*(a^2 - b^2*c)*(c + d*x)^2)/(4*b^4) + (2*a*(c + d*x)^(5/2))/(5*b^3) - (c + d*x)^3/(6*b^2) - (a*(a^2 - b^2*c)^3)/(b^8*(a + b*Sqrt[c + d*x])) - ((a^2 - b^2*c)^2*(7*a^2 - b^2*c)*Log[a + b*Sqrt[c + d*x]])/b^8))/d^4
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 = 0.05 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{3} b^{5}}{6}+\frac {2 a \left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}+\frac {3 b^{5} c \left (d x +c \right )^{2}}{4}-\frac {3 a^{2} b^{3} \left (d x +c \right )^{2}}{4}-2 a \,b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 b^{5} c^{2} \left (d x +c \right )}{2}+\frac {4 a^{3} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {9 a^{2} b^{3} c \left (d x +c \right )}{2}+6 a \,c^{2} b^{4} \sqrt {d x +c}-\frac {5 a^{4} b \left (d x +c \right )}{2}-12 a^{3} c \,b^{2} \sqrt {d x +c}+6 a^{5} \sqrt {d x +c}\right )}{b^{7}}+\frac {2 \left (-b^{6} c^{3}+9 a^{2} b^{4} c^{2}-15 a^{4} b^{2} c +7 a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}+\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right )}{b^{8} \left (a +b \sqrt {d x +c}\right )}}{d^{4}}\) | \(277\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{3} b^{5}}{6}+\frac {2 a \left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}+\frac {3 b^{5} c \left (d x +c \right )^{2}}{4}-\frac {3 a^{2} b^{3} \left (d x +c \right )^{2}}{4}-2 a \,b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 b^{5} c^{2} \left (d x +c \right )}{2}+\frac {4 a^{3} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {9 a^{2} b^{3} c \left (d x +c \right )}{2}+6 a \,c^{2} b^{4} \sqrt {d x +c}-\frac {5 a^{4} b \left (d x +c \right )}{2}-12 a^{3} c \,b^{2} \sqrt {d x +c}+6 a^{5} \sqrt {d x +c}\right )}{b^{7}}+\frac {2 \left (-b^{6} c^{3}+9 a^{2} b^{4} c^{2}-15 a^{4} b^{2} c +7 a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}+\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right )}{b^{8} \left (a +b \sqrt {d x +c}\right )}}{d^{4}}\) | \(277\) |
Input:
int(x^3/(a+b*(d*x+c)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
2/d^4*(-1/b^7*(-1/6*(d*x+c)^3*b^5+2/5*a*(d*x+c)^(5/2)*b^4+3/4*b^5*c*(d*x+c )^2-3/4*a^2*b^3*(d*x+c)^2-2*a*b^4*c*(d*x+c)^(3/2)-3/2*b^5*c^2*(d*x+c)+4/3* a^3*b^2*(d*x+c)^(3/2)+9/2*a^2*b^3*c*(d*x+c)+6*a*c^2*b^4*(d*x+c)^(1/2)-5/2* a^4*b*(d*x+c)-12*a^3*c*b^2*(d*x+c)^(1/2)+6*a^5*(d*x+c)^(1/2))+1/b^8*(-b^6* c^3+9*a^2*b^4*c^2-15*a^4*b^2*c+7*a^6)*ln(a+b*(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/(a+b*(d*x+c)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.63 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {10 \, b^{8} d^{4} x^{4} + 55 \, b^{8} c^{4} - 220 \, a^{2} b^{6} c^{3} + 195 \, a^{4} b^{4} c^{2} + 30 \, a^{6} b^{2} c - 60 \, a^{8} - 5 \, {\left (b^{8} c - 7 \, a^{2} b^{6}\right )} d^{3} x^{3} + 15 \, {\left (b^{8} c^{2} - 8 \, a^{2} b^{6} c + 7 \, a^{4} b^{4}\right )} d^{2} x^{2} + 5 \, {\left (17 \, b^{8} c^{3} - 87 \, a^{2} b^{6} c^{2} + 96 \, a^{4} b^{4} c - 30 \, a^{6} b^{2}\right )} d x - 60 \, {\left (b^{8} c^{4} - 10 \, a^{2} b^{6} c^{3} + 24 \, a^{4} b^{4} c^{2} - 22 \, a^{6} b^{2} c + 7 \, a^{8} + {\left (b^{8} c^{3} - 9 \, a^{2} b^{6} c^{2} + 15 \, a^{4} b^{4} c - 7 \, a^{6} b^{2}\right )} d x\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (6 \, a b^{7} d^{3} x^{3} + 81 \, a b^{7} c^{3} - 271 \, a^{3} b^{5} c^{2} + 295 \, a^{5} b^{3} c - 105 \, a^{7} b - 2 \, {\left (6 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} x^{2} + 2 \, {\left (24 \, a b^{7} c^{2} - 61 \, a^{3} b^{5} c + 35 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}}{30 \, {\left (b^{10} d^{5} x + {\left (b^{10} c - a^{2} b^{8}\right )} d^{4}\right )}} \] Input:
integrate(x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="fricas")
Output:
1/30*(10*b^8*d^4*x^4 + 55*b^8*c^4 - 220*a^2*b^6*c^3 + 195*a^4*b^4*c^2 + 30 *a^6*b^2*c - 60*a^8 - 5*(b^8*c - 7*a^2*b^6)*d^3*x^3 + 15*(b^8*c^2 - 8*a^2* b^6*c + 7*a^4*b^4)*d^2*x^2 + 5*(17*b^8*c^3 - 87*a^2*b^6*c^2 + 96*a^4*b^4*c - 30*a^6*b^2)*d*x - 60*(b^8*c^4 - 10*a^2*b^6*c^3 + 24*a^4*b^4*c^2 - 22*a^ 6*b^2*c + 7*a^8 + (b^8*c^3 - 9*a^2*b^6*c^2 + 15*a^4*b^4*c - 7*a^6*b^2)*d*x )*log(sqrt(d*x + c)*b + a) - 4*(6*a*b^7*d^3*x^3 + 81*a*b^7*c^3 - 271*a^3*b ^5*c^2 + 295*a^5*b^3*c - 105*a^7*b - 2*(6*a*b^7*c - 7*a^3*b^5)*d^2*x^2 + 2 *(24*a*b^7*c^2 - 61*a^3*b^5*c + 35*a^5*b^3)*d*x)*sqrt(d*x + c))/(b^10*d^5* x + (b^10*c - a^2*b^8)*d^4)
Time = 6.80 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.11 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\begin {cases} \frac {2 \left (- \frac {2 a \left (c + d x\right )^{\frac {5}{2}}}{5 b^{3}} - \frac {a \left (a^{2} - b^{2} c\right )^{3} \left (\begin {cases} \frac {\sqrt {c + d x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {1}{b \left (a + b \sqrt {c + d x}\right )} & \text {otherwise} \end {cases}\right )}{b^{7}} + \frac {\left (c + d x\right )^{3}}{6 b^{2}} + \frac {\left (3 a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{2}}{4 b^{4}} + \frac {\left (- 4 a^{3} + 6 a b^{2} c\right ) \left (c + d x\right )^{\frac {3}{2}}}{3 b^{5}} + \frac {\left (c + d x\right ) \left (5 a^{4} - 9 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{2 b^{6}} + \frac {\left (a^{2} - b^{2} c\right )^{2} \cdot \left (7 a^{2} - b^{2} c\right ) \left (\begin {cases} \frac {\sqrt {c + d x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {c + d x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{7}} + \frac {\sqrt {c + d x} \left (- 6 a^{5} + 12 a^{3} b^{2} c - 6 a b^{4} c^{2}\right )}{b^{7}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4}}{4 \left (a + b \sqrt {c}\right )^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(x**3/(a+b*(d*x+c)**(1/2))**2,x)
Output:
Piecewise((2*(-2*a*(c + d*x)**(5/2)/(5*b**3) - a*(a**2 - b**2*c)**3*Piecew ise((sqrt(c + d*x)/a**2, Eq(b, 0)), (-1/(b*(a + b*sqrt(c + d*x))), True))/ b**7 + (c + d*x)**3/(6*b**2) + (3*a**2 - 3*b**2*c)*(c + d*x)**2/(4*b**4) + (-4*a**3 + 6*a*b**2*c)*(c + d*x)**(3/2)/(3*b**5) + (c + d*x)*(5*a**4 - 9* a**2*b**2*c + 3*b**4*c**2)/(2*b**6) + (a**2 - b**2*c)**2*(7*a**2 - b**2*c) *Piecewise((sqrt(c + d*x)/a, Eq(b, 0)), (log(a + b*sqrt(c + d*x))/b, True) )/b**7 + sqrt(c + d*x)*(-6*a**5 + 12*a**3*b**2*c - 6*a*b**4*c**2)/b**7)/d* *4, Ne(d, 0)), (x**4/(4*(a + b*sqrt(c))**2), True))
Time = 0.03 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.05 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=-\frac {\frac {60 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{\sqrt {d x + c} b^{9} + a b^{8}} - \frac {10 \, {\left (d x + c\right )}^{3} b^{5} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{4} - 45 \, {\left (b^{5} c - a^{2} b^{3}\right )} {\left (d x + c\right )}^{2} + 40 \, {\left (3 \, a b^{4} c - 2 \, a^{3} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 30 \, {\left (3 \, b^{5} c^{2} - 9 \, a^{2} b^{3} c + 5 \, a^{4} b\right )} {\left (d x + c\right )} - 360 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt {d x + c}}{b^{7}} + \frac {60 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8}}}{30 \, d^{4}} \] Input:
integrate(x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="maxima")
Output:
-1/30*(60*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)/(sqrt(d*x + c)*b ^9 + a*b^8) - (10*(d*x + c)^3*b^5 - 24*(d*x + c)^(5/2)*a*b^4 - 45*(b^5*c - a^2*b^3)*(d*x + c)^2 + 40*(3*a*b^4*c - 2*a^3*b^2)*(d*x + c)^(3/2) + 30*(3 *b^5*c^2 - 9*a^2*b^3*c + 5*a^4*b)*(d*x + c) - 360*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*sqrt(d*x + c))/b^7 + 60*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*log(sqrt(d*x + c)*b + a)/b^8)/d^4
Time = 0.32 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.35 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=-\frac {2 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} - \frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{{\left (\sqrt {d x + c} b + a\right )} b^{8} d^{4}} + \frac {10 \, {\left (d x + c\right )}^{3} b^{10} d^{20} - 45 \, {\left (d x + c\right )}^{2} b^{10} c d^{20} + 90 \, {\left (d x + c\right )} b^{10} c^{2} d^{20} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{9} d^{20} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{9} c d^{20} - 360 \, \sqrt {d x + c} a b^{9} c^{2} d^{20} + 45 \, {\left (d x + c\right )}^{2} a^{2} b^{8} d^{20} - 270 \, {\left (d x + c\right )} a^{2} b^{8} c d^{20} - 80 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} d^{20} + 720 \, \sqrt {d x + c} a^{3} b^{7} c d^{20} + 150 \, {\left (d x + c\right )} a^{4} b^{6} d^{20} - 360 \, \sqrt {d x + c} a^{5} b^{5} d^{20}}{30 \, b^{12} d^{24}} \] Input:
integrate(x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="giac")
Output:
-2*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*log(abs(sqrt(d*x + c)* b + a))/(b^8*d^4) - 2*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)/((sq rt(d*x + c)*b + a)*b^8*d^4) + 1/30*(10*(d*x + c)^3*b^10*d^20 - 45*(d*x + c )^2*b^10*c*d^20 + 90*(d*x + c)*b^10*c^2*d^20 - 24*(d*x + c)^(5/2)*a*b^9*d^ 20 + 120*(d*x + c)^(3/2)*a*b^9*c*d^20 - 360*sqrt(d*x + c)*a*b^9*c^2*d^20 + 45*(d*x + c)^2*a^2*b^8*d^20 - 270*(d*x + c)*a^2*b^8*c*d^20 - 80*(d*x + c) ^(3/2)*a^3*b^7*d^20 + 720*sqrt(d*x + c)*a^3*b^7*c*d^20 + 150*(d*x + c)*a^4 *b^6*d^20 - 360*sqrt(d*x + c)*a^5*b^5*d^20)/(b^12*d^24)
Time = 0.09 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.92 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\left (\frac {4\,a^3}{3\,b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{3\,b}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {3\,c}{2\,b^2\,d^4}-\frac {3\,a^2}{2\,b^4\,d^4}\right )\,{\left (c+d\,x\right )}^2-\left (\frac {2\,a\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {6\,c^2}{b^2\,d^4}\right )}{b}+\frac {a^2\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b^2}\right )\,\sqrt {c+d\,x}+\frac {{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}+\frac {2\,\left (a^7-3\,a^5\,b^2\,c+3\,a^3\,b^4\,c^2-a\,b^6\,c^3\right )}{b\,\left (b^8\,d^4\,\sqrt {c+d\,x}+a\,b^7\,d^4\right )}+d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{2\,b^2}-\frac {a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {3\,c^2}{b^2\,d^4}\right )+\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (14\,a^6-30\,a^4\,b^2\,c+18\,a^2\,b^4\,c^2-2\,b^6\,c^3\right )}{b^8\,d^4}-\frac {4\,a\,{\left (c+d\,x\right )}^{5/2}}{5\,b^3\,d^4} \] Input:
int(x^3/(a + b*(c + d*x)^(1/2))^2,x)
Output:
((4*a^3)/(3*b^5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/(3*b))* (c + d*x)^(3/2) - ((3*c)/(2*b^2*d^4) - (3*a^2)/(2*b^4*d^4))*(c + d*x)^2 - ((2*a*((a^2*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b^2 - (2*a*((4*a^3)/(b^ 5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b))/b + (6*c^2)/(b^2* d^4)))/b + (a^2*((4*a^3)/(b^5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4* d^4)))/b))/b^2)*(c + d*x)^(1/2) + (c + d*x)^3/(3*b^2*d^4) + (2*(a^7 - 3*a^ 5*b^2*c - a*b^6*c^3 + 3*a^3*b^4*c^2))/(b*(b^8*d^4*(c + d*x)^(1/2) + a*b^7* d^4)) + d*x*((a^2*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/(2*b^2) - (a*((4* a^3)/(b^5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b))/b + (3*c^ 2)/(b^2*d^4)) + (log(a + b*(c + d*x)^(1/2))*(14*a^6 - 2*b^6*c^3 - 30*a^4*b ^2*c + 18*a^2*b^4*c^2))/(b^8*d^4) - (4*a*(c + d*x)^(5/2))/(5*b^3*d^4)
Time = 0.28 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.93 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {420 \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a^{6} b -60 \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) b^{7} c^{3}+540 \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a^{2} b^{5} c^{2}-900 \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a^{4} b^{3} c +970 \sqrt {d x +c}\, a^{4} b^{3} c -669 \sqrt {d x +c}\, a^{2} b^{5} c^{2}+10 \sqrt {d x +c}\, b^{7} d^{3} x^{3}-900 \,\mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a^{5} b^{2} c +540 \,\mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a^{3} b^{4} c^{2}-60 \,\mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a \,b^{6} c^{3}-210 a^{5} b^{2} d x -35 a^{3} b^{4} d^{2} x^{2}-14 a \,b^{6} d^{3} x^{3}-420 \sqrt {d x +c}\, a^{6} b +115 \sqrt {d x +c}\, b^{7} c^{3}-210 a^{5} b^{2} c +415 a^{3} b^{4} c^{2}-209 a \,b^{6} c^{3}+70 \sqrt {d x +c}\, a^{4} b^{3} d x +21 \sqrt {d x +c}\, a^{2} b^{5} d^{2} x^{2}+30 \sqrt {d x +c}\, b^{7} c^{2} d x -15 \sqrt {d x +c}\, b^{7} c \,d^{2} x^{2}+380 a^{3} b^{4} c d x -162 a \,b^{6} c^{2} d x +33 a \,b^{6} c \,d^{2} x^{2}+420 \,\mathrm {log}\left (\sqrt {d x +c}\, b +a \right ) a^{7}-108 \sqrt {d x +c}\, a^{2} b^{5} c d x}{30 b^{8} d^{4} \left (\sqrt {d x +c}\, b +a \right )} \] Input:
int(x^3/(a+b*(d*x+c)^(1/2))^2,x)
Output:
(420*sqrt(c + d*x)*log(sqrt(c + d*x)*b + a)*a**6*b - 900*sqrt(c + d*x)*log (sqrt(c + d*x)*b + a)*a**4*b**3*c + 540*sqrt(c + d*x)*log(sqrt(c + d*x)*b + a)*a**2*b**5*c**2 - 60*sqrt(c + d*x)*log(sqrt(c + d*x)*b + a)*b**7*c**3 - 420*sqrt(c + d*x)*a**6*b + 970*sqrt(c + d*x)*a**4*b**3*c + 70*sqrt(c + d *x)*a**4*b**3*d*x - 669*sqrt(c + d*x)*a**2*b**5*c**2 - 108*sqrt(c + d*x)*a **2*b**5*c*d*x + 21*sqrt(c + d*x)*a**2*b**5*d**2*x**2 + 115*sqrt(c + d*x)* b**7*c**3 + 30*sqrt(c + d*x)*b**7*c**2*d*x - 15*sqrt(c + d*x)*b**7*c*d**2* x**2 + 10*sqrt(c + d*x)*b**7*d**3*x**3 + 420*log(sqrt(c + d*x)*b + a)*a**7 - 900*log(sqrt(c + d*x)*b + a)*a**5*b**2*c + 540*log(sqrt(c + d*x)*b + a) *a**3*b**4*c**2 - 60*log(sqrt(c + d*x)*b + a)*a*b**6*c**3 - 210*a**5*b**2* c - 210*a**5*b**2*d*x + 415*a**3*b**4*c**2 + 380*a**3*b**4*c*d*x - 35*a**3 *b**4*d**2*x**2 - 209*a*b**6*c**3 - 162*a*b**6*c**2*d*x + 33*a*b**6*c*d**2 *x**2 - 14*a*b**6*d**3*x**3)/(30*b**8*d**4*(sqrt(c + d*x)*b + a))