Integrand size = 26, antiderivative size = 243 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 x \sqrt {c+d x}}{b d \sqrt {a x+b x^2}}-\frac {2 \sqrt {a} (b c-2 a d) \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{b^{3/2} d (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}-\frac {2 a^{3/2} \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:
2*x*(d*x+c)^(1/2)/b/d/(b*x^2+a*x)^(1/2)-2*a^(1/2)*(-2*a*d+b*c)*x^(1/2)*(d* x+c)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^( 1/2))/b^(3/2)/d/(-a*d+b*c)/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/2)-2 *a^(3/2)*x^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x^(1/2)/a^(1 /2)),(1-a*d/b/c)^(1/2))/b^(3/2)/(-a*d+b*c)/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b* x^2+a*x)^(1/2)
Result contains complex when optimal does not.
Time = 11.25 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.99 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 (c+d x) \left (-2 a^2 d+b^2 c x+a b (c-d x)\right )-2 i \sqrt {\frac {a}{b}} b d (-b c+2 a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+4 i \sqrt {\frac {a}{b}} b d (-b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{b^2 d (b c-a d) \sqrt {x (a+b x)} \sqrt {c+d x}} \] Input:
Integrate[x^3/(Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
Output:
(2*(c + d*x)*(-2*a^2*d + b^2*c*x + a*b*(c - d*x)) - (2*I)*Sqrt[a/b]*b*d*(- (b*c) + 2*a*d)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticE[I*Arc Sinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] + (4*I)*Sqrt[a/b]*b*d*(-(b*c) + a*d) *Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[a/b] /Sqrt[x]], (b*c)/(a*d)])/(b^2*d*(b*c - a*d)*Sqrt[x*(a + b*x)]*Sqrt[c + d*x ])
Time = 0.64 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1261, 109, 27, 176, 122, 120, 127, 126}
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.
\(\displaystyle \int \frac {x^3}{\left (a x+b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 1261 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \int \frac {x^{3/2}}{(a+b x)^{3/2} \sqrt {c+d x}}dx}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {2 \int \frac {a c-(b c-2 a d) x}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\int \frac {a c-(b c-2 a d) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {c (b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {(b c-2 a d) \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {c (b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {c (b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[x^3/(Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
Output:
(x^(3/2)*(a + b*x)^(3/2)*((2*a*Sqrt[x]*Sqrt[c + d*x])/(b*(b*c - a*d)*Sqrt[ a + b*x]) - ((-2*Sqrt[-a]*(b*c - 2*a*d)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*El lipticE[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[ a + b*x]*Sqrt[1 + (d*x)/c]) + (2*Sqrt[-a]*c*(b*c - a*d)*Sqrt[1 + (b*x)/a]* Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c )])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[c + d*x]))/(b*(b*c - a*d))))/(a*x + b*x^ 2)^(3/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 1.54 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.66
method | result | size |
elliptic | \(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (b d \,x^{2}+c b x \right ) a}{\left (a d -b c \right ) b^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d \,x^{2}+c b x \right )}}+\frac {2 c^{2} a \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b \left (a d -b c \right ) d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}+\frac {2 \left (\frac {1}{b}+\frac {a d}{b \left (a d -b c \right )}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\) | \(404\) |
default | \(-\frac {2 \left (2 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}-2 a \,c^{2} \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b d -2 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -\sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3}+a b \,d^{3} x^{2}+a b c \,d^{2} x \right ) \sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}{d^{2} b^{2} \left (a d -b c \right ) x \left (b d \,x^{2}+a d x +c b x +a c \right )}\) | \(453\) |
Input:
int(x^3/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
(x*(b*x+a)*(d*x+c))^(1/2)/(x*(b*x+a))^(1/2)/(d*x+c)^(1/2)*(-2*(b*d*x^2+b*c *x)/(a*d-b*c)/b^2*a/((x+a/b)*(b*d*x^2+b*c*x))^(1/2)+2/b*c^2/(a*d-b*c)*a/d* ((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*x^3+a *d*x^2+b*c*x^2+a*c*x)^(1/2)*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b) )^(1/2))+2*(1/b+a/b*d/(a*d-b*c))*c/d*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/ b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)*((-c/d+a/ b)*EllipticE(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))-a/b*EllipticF((( x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (216) = 432\).
Time = 0.12 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.81 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x^{2} + a x} \sqrt {d x + c} a b^{2} d^{2} - {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 2 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) - 3 \, {\left (a b^{2} c d - 2 \, a^{2} b d^{2} + {\left (b^{3} c d - 2 \, a b^{2} d^{2}\right )} x\right )} \sqrt {b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x\right )}} \] Input:
integrate(x^3/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
2/3*(3*sqrt(b*x^2 + a*x)*sqrt(d*x + c)*a*b^2*d^2 - (a*b^2*c^2 + 2*a^2*b*c* d - 2*a^3*d^2 + (b^3*c^2 + 2*a*b^2*c*d - 2*a^2*b*d^2)*x)*sqrt(b*d)*weierst rassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b* c + a*d)/(b*d)) - 3*(a*b^2*c*d - 2*a^2*b*d^2 + (b^3*c*d - 2*a*b^2*d^2)*x)* sqrt(b*d)*weierstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/ 27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), weie rstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3* c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d))))/(a*b^4*c*d^2 - a^2*b^3*d^3 + (b^5*c*d^2 - a*b^4*d^3)* x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x**3/(d*x+c)**(1/2)/(b*x**2+a*x)**(3/2),x)
Output:
Integral(x**3/((x*(a + b*x))**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^3/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^3/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)), x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^3/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(x^3/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (b\,x^2+a\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(x^3/((a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int(x^3/((a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^3}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \] Input:
int(x^3/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x)
Output:
int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c + a**2*d*x + 2*a*b*c*x + 2*a*b*d*x**2 + b**2*c*x**2 + b**2*d*x**3),x)