Integrand size = 26, antiderivative size = 318 \[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {4 (b c+2 a d) x \sqrt {c+d x}}{3 b^2 d^2 \sqrt {a x+b x^2}}+\frac {2 x^2 \sqrt {c+d x}}{3 b d \sqrt {a x+b x^2}}+\frac {2 \sqrt {a} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}-\frac {2 a^{3/2} (b c-4 a d) \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} d (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:
-4/3*(2*a*d+b*c)*x*(d*x+c)^(1/2)/b^2/d^2/(b*x^2+a*x)^(1/2)+2/3*x^2*(d*x+c) ^(1/2)/b/d/(b*x^2+a*x)^(1/2)+2/3*a^(1/2)*(-8*a^2*d^2+3*a*b*c*d+2*b^2*c^2)* 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^(5/2)/d^2/(-a*d+b*c)/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2 +a*x)^(1/2)-2/3*a^(3/2)*(-4*a*d+b*c)*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^(5/2)/d/(-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 = 14.78 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.97 \[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 (c+d x) \left (-8 a^3 d^2+a^2 b d (3 c-4 d x)+b^3 c x (2 c-d x)+a b^2 \left (2 c^2+2 c d x+d^2 x^2\right )\right )-2 i \sqrt {\frac {a}{b}} b d \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d \left (-b^2 c^2-7 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{3 b^3 d^2 (-b c+a d) \sqrt {x (a+b x)} \sqrt {c+d x}} \] Input:
Integrate[x^4/(Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
Output:
(2*(c + d*x)*(-8*a^3*d^2 + a^2*b*d*(3*c - 4*d*x) + b^3*c*x*(2*c - d*x) + a *b^2*(2*c^2 + 2*c*d*x + d^2*x^2)) - (2*I)*Sqrt[a/b]*b*d*(-2*b^2*c^2 - 3*a* b*c*d + 8*a^2*d^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticE[I *ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] + (2*I)*Sqrt[a/b]*b*d*(-(b^2*c^2 ) - 7*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*Ell ipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)])/(3*b^3*d^2*(-(b*c) + a* d)*Sqrt[x*(a + b*x)]*Sqrt[c + d*x])
Time = 0.80 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1261, 109, 27, 171, 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.
| \(\displaystyle \int \frac {x^4}{\left (a x+b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \int \frac {x^{5/2}}{(a+b x)^{3/2} \sqrt {c+d x}}dx}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {2 \int \frac {\sqrt {x} (3 a c-(b c-4 a d) x)}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\int \frac {\sqrt {x} (3 a c-(b c-4 a d) x)}{\sqrt {a+b x} \sqrt {c+d x}}dx}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {2 \int \frac {a c (b c-4 a d)+\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\int \frac {a c (b c-4 a d)+\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {2 c (b c-a d) (2 a d+b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 c (b c-a d) (2 a d+b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 c (b c-a d) (2 a d+b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) (2 a d+b c) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (\frac {2 a x^{3/2} \sqrt {c+d x}}{b \sqrt {a+b x} (b c-a d)}-\frac {\frac {\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {c+d x}}}{3 b d}-\frac {2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x} (b c-4 a d)}{3 b d}}{b (b c-a d)}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[x^4/(Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
Output:
(x^(3/2)*(a + b*x)^(3/2)*((2*a*x^(3/2)*Sqrt[c + d*x])/(b*(b*c - a*d)*Sqrt[ a + b*x]) - ((-2*(b*c - 4*a*d)*Sqrt[x]*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*b*d ) + ((2*Sqrt[-a]*(2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*Sqrt[1 + (b*x)/a]*Sqr t[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sq rt[b]*d*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (4*Sqrt[-a]*c*(b*c - a*d)*(b*c + 2*a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqr t[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[c + d*x]))/(3 *b*d))/(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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.66 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.49
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (\frac {2 \left (b d \,x^{2}+c b x \right ) a^{2}}{\left (a d -b c \right ) b^{3} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d \,x^{2}+c b x \right )}}+\frac {2 \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{3 b^{2} d}+\frac {2 \left (-\frac {c \,a^{2}}{b^{2} \left (a d -b c \right )}-\frac {a c}{3 b^{2} d}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}+\frac {2 \left (-\frac {a}{b^{2}}-\frac {d \,a^{2}}{b^{2} \left (a d -b c \right )}-\frac {2 \left (a d +b c \right )}{3 b^{2} d}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\) | \(474\) |
| default | \(\frac {2 \left (8 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}-7 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}-\sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -8 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}+11 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}-\sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -2 \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{3} c^{4}+a \,b^{2} d^{4} x^{3}-b^{3} c \,d^{3} x^{3}+4 a^{2} b \,d^{4} x^{2}-b^{3} c^{2} d^{2} x^{2}+4 a^{2} c \,d^{3} x b -a \,b^{2} c^{2} d^{2} x \right ) \sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}{3 b^{3} d^{3} \left (a d -b c \right ) x \left (b d \,x^{2}+a d x +c b x +a c \right )}\) | \(671\) |
Input:
int(x^4/(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^3*a^2/((x+a/b)*(b*d*x^2+b*c*x))^(1/2)+2/3/b^2/d*(b*d*x^3+a* d*x^2+b*c*x^2+a*c*x)^(1/2)+2*(-1/b^2*c/(a*d-b*c)*a^2-1/3/b^2/d*a*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)*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^ (1/2))+2*(-a/b^2-d/b^2*a^2/(a*d-b*c)-2/3/b^2/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))))
Time = 0.13 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.67 \[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left (2 \, a b^{3} c^{3} + 2 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 8 \, a^{4} d^{3} + {\left (2 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 8 \, a^{3} b d^{3}\right )} x\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (2 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - 8 \, a^{3} b d^{3} + {\left (2 \, b^{4} c^{2} d + 3 \, a b^{3} c d^{2} - 8 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, {\left (a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3} + {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {d x + c}\right )}}{9 \, {\left (a b^{5} c d^{3} - a^{2} b^{4} d^{4} + {\left (b^{6} c d^{3} - a b^{5} d^{4}\right )} x\right )}} \] Input:
integrate(x^4/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
2/9*((2*a*b^3*c^3 + 2*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 8*a^4*d^3 + (2*b^4*c ^3 + 2*a*b^3*c^2*d + 7*a^2*b^2*c*d^2 - 8*a^3*b*d^3)*x)*sqrt(b*d)*weierstra ssPInverse(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*(2*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - 8*a^3*b*d^3 + (2*b^4* c^2*d + 3*a*b^3*c*d^2 - 8*a^2*b^2*d^3)*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), weierstrassPInverse(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^3*c *d^2 - 4*a^2*b^2*d^3 + (b^4*c*d^2 - a*b^3*d^3)*x)*sqrt(b*x^2 + a*x)*sqrt(d *x + c))/(a*b^5*c*d^3 - a^2*b^4*d^4 + (b^6*c*d^3 - a*b^5*d^4)*x)
\[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x**4/(d*x+c)**(1/2)/(b*x**2+a*x)**(3/2),x)
Output:
Integral(x**4/((x*(a + b*x))**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^4/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^4/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)), x)
\[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^4/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(x^4/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (b\,x^2+a\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(x^4/((a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int(x^4/((a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^4}{\sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {-6 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, c +4 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, d x +3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x +2 \sqrt {x}\, a b c x +2 \sqrt {x}\, a b d \,x^{2}+\sqrt {x}\, b^{2} c \,x^{2}+\sqrt {x}\, b^{2} d \,x^{3}}d x \right ) a^{2} c^{2}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x +2 \sqrt {x}\, a b c x +2 \sqrt {x}\, a b d \,x^{2}+\sqrt {x}\, b^{2} c \,x^{2}+\sqrt {x}\, b^{2} d \,x^{3}}d x \right ) a b \,c^{2} x -8 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) a^{2} d^{2}-\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) a b c d -8 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) a b \,d^{2} x -\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{b^{2} d \,x^{3}+2 a b d \,x^{2}+b^{2} c \,x^{2}+a^{2} d x +2 a b c x +a^{2} c}d x \right ) b^{2} c d x}{6 b \,d^{2} \left (b x +a \right )} \] Input:
int(x^4/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x)
Output:
( - 6*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*c + 4*sqrt(x)*sqrt(c + d*x)*sqrt (a + b*x)*d*x + 3*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c + sqrt (x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x **2 + sqrt(x)*b**2*d*x**3),x)*a**2*c**2 + 3*int((sqrt(c + d*x)*sqrt(a + b* x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b *d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b*c**2*x - 8*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)*a**2*d**2 - 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)*a*b*c*d - 8*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)*a*b*d**2*x - 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)*b**2*c*d*x)/(6*b*d**2*(a + b*x))