Integrand size = 28, antiderivative size = 224 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(d+e x)^{3/2}}{a+b x+c x^2}-\frac {3 e \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {3 e \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \] Output:
-(e*x+d)^(3/2)/(c*x^2+b*x+a)-3/2*e*(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)* arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/ 2))*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/2)+3/2*e*(2*c*d-(b+(-4*a*c+b^2)^(1/2)) *e)^(1/2)*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/ 2))*e)^(1/2))*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/2)
Result contains complex when optimal does not.
Time = 4.88 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.42 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(d+e x)^{3/2}}{a+x (b+c x)}+\frac {2 \sqrt {2} e \left (4 c d+\left (-2 b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {e \left (-10 i c d+\left (5 i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {e \left (10 i c d+\left (-5 i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\frac {2 \sqrt {2} e \left (-4 c d+\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Input:
Integrate[((b + 2*c*x)*(d + e*x)^(3/2))/(a + b*x + c*x^2)^2,x]
Output:
-((d + e*x)^(3/2)/(a + x*(b + c*x))) + (2*Sqrt[2]*e*(4*c*d + (-2*b + Sqrt[ b^2 - 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqr t[b^2 - 4*a*c])*e]) - (e*((-10*I)*c*d + ((5*I)*b + Sqrt[-b^2 + 4*a*c])*e)* ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4 *a*c]*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[- b^2 + 4*a*c])*e]) - (e*((10*I)*c*d + ((-5*I)*b + Sqrt[-b^2 + 4*a*c])*e)*Ar cTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a *c]*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^ 2 + 4*a*c])*e]) + (2*Sqrt[2]*e*(-4*c*d + (2*b + Sqrt[b^2 - 4*a*c])*e)*ArcT an[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e ]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e])
Time = 0.50 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1222, 1148, 1450, 221}
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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle \frac {3}{2} e \int \frac {\sqrt {d+e x}}{c x^2+b x+a}dx-\frac {(d+e x)^{3/2}}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1148 |
| \(\displaystyle 3 e^2 \int \frac {d+e x}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}-\frac {(d+e x)^{3/2}}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1450 |
| \(\displaystyle 3 e^2 \left (\frac {1}{2} \left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left (1-\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )-\frac {(d+e x)^{3/2}}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 3 e^2 \left (-\frac {\left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (1-\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )-\frac {(d+e x)^{3/2}}{a+b x+c x^2}\) |
Input:
Int[((b + 2*c*x)*(d + e*x)^(3/2))/(a + b*x + c*x^2)^2,x]
Output:
-((d + e*x)^(3/2)/(a + b*x + c*x^2)) + 3*e^2*(-(((1 + (2*c*d - b*e)/(Sqrt[ b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a *c])*e])) - ((1 - (2*c*d - b*e)/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sq rt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sq rt[c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[2*e Subst[Int[x^2/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c *x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) Int[(d + e*x)^(m - 1)* (a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1) Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Time = 1.83 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(2 e^{2} \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{2 \left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )}+6 c \left (-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )\) | \(309\) |
| default | \(2 e^{2} \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{2 \left (c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}\right )}+6 c \left (-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\right )\) | \(309\) |
| pseudoelliptic | \(\frac {-3 \left (c \,x^{2}+b x +a \right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, e^{2} \sqrt {2}\, \left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+3 \left (\sqrt {2}\, e^{2} \left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \left (c \,x^{2}+b x +a \right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\frac {2 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{3}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (2 c \,x^{2}+2 b x +2 a \right )}\) | \(372\) |
Input:
int((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
2*e^2*(-1/2*(e*x+d)^(3/2)/(c*(e*x+d)^2+b*e*(e*x+d)-2*c*d*(e*x+d)+a*e^2-b*d *e+c*d^2)+6*c*(-1/8*(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))/c/(-e^2*(4*a*c-b ^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh ((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))+ 1/8*(b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2 )/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^ (1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (182) = 364\).
Time = 0.09 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.70 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
-1/2*(3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^ 2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 + 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt ((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^ 2*c - 4*a*c^2))) - 3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))*log(2 7*sqrt(e*x + d)*e^4 - 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) - 3*sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d* e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4* a*c^2))*log(27*sqrt(e*x + d)*e^4 + 27*sqrt(1/2)*sqrt(e^6/(b^2*c^2 - 4*a*c^ 3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^ 3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) + 3*sqrt(1/2)*(c*x^2 + b*x + a) *sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2*c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2) )/(b^2*c - 4*a*c^2))*log(27*sqrt(e*x + d)*e^4 - 27*sqrt(1/2)*sqrt(e^6/(b^2 *c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/(b^2 *c^2 - 4*a*c^3))*(b^2*c - 4*a*c^2))/(b^2*c - 4*a*c^2))) + 2*(e*x + d)^(3/2 ))/(c*x^2 + b*x + a)
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((2*c*x+b)*(e*x+d)**(3/2)/(c*x**2+b*x+a)**2,x)
Output:
Timed out
\[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}} \,d x } \] Input:
integrate((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
integrate((2*c*x + b)*(e*x + d)^(3/2)/(c*x^2 + b*x + a)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (182) = 364\).
Time = 0.34 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.25 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}{{\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e + a e^{2}} + \frac {3 \, {\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{4} - {\left (4 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} b c d e + \sqrt {b^{2} - 4 \, a c} a c e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {3 \, {\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{4} - {\left (4 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3} + b^{2} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} b c d e + \sqrt {b^{2} - 4 \, a c} a c e^{2}\right )} {\left | c \right |} {\left | e \right |}} \] Input:
integrate((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
-(e*x + d)^(3/2)*e^2/((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)* b*e - b*d*e + a*e^2) + 3/8*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)* e)*(b^2 - 4*a*c)*e^4 - (4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(-4*c^2 *d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sq rt(-(2*c*d - b*e + sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c) )/((sqrt(b^2 - 4*a*c)*c^2*d^2 - sqrt(b^2 - 4*a*c)*b*c*d*e + sqrt(b^2 - 4*a *c)*a*c*e^2)*abs(c)*abs(e)) - 3/8*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a *c)*c)*e)*(b^2 - 4*a*c)*e^4 - (4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt (-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2) *c))/c))/((sqrt(b^2 - 4*a*c)*c^2*d^2 - sqrt(b^2 - 4*a*c)*b*c*d*e + sqrt(b^ 2 - 4*a*c)*a*c*e^2)*abs(c)*abs(e))
Time = 0.48 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.75 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-18\,b^2\,c\,e^6+36\,b\,c^2\,d\,e^5-36\,c^3\,d^2\,e^4+36\,a\,c^2\,e^6\right )+\frac {9\,\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (b^3\,e^3+e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,e^3+8\,a\,c^2\,d\,e^2-2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {-\frac {9\,\left (b^3\,e^3+e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,e^3+8\,a\,c^2\,d\,e^2-2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{54\,c^2\,d^2\,e^6-54\,b\,c\,d\,e^7+54\,a\,c\,e^8}\right )\,\sqrt {-\frac {9\,\left (b^3\,e^3+e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,e^3+8\,a\,c^2\,d\,e^2-2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-18\,b^2\,c\,e^6+36\,b\,c^2\,d\,e^5-36\,c^3\,d^2\,e^4+36\,a\,c^2\,e^6\right )-\frac {9\,\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e^3+4\,a\,b\,c\,e^3-8\,a\,c^2\,d\,e^2+2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {\frac {9\,\left (e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e^3+4\,a\,b\,c\,e^3-8\,a\,c^2\,d\,e^2+2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{54\,c^2\,d^2\,e^6-54\,b\,c\,d\,e^7+54\,a\,c\,e^8}\right )\,\sqrt {\frac {9\,\left (e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e^3+4\,a\,b\,c\,e^3-8\,a\,c^2\,d\,e^2+2\,b^2\,c\,d\,e^2\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-\frac {e^2\,{\left (d+e\,x\right )}^{3/2}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )+c\,{\left (d+e\,x\right )}^2+a\,e^2+c\,d^2-b\,d\,e} \] Input:
int(((b + 2*c*x)*(d + e*x)^(3/2))/(a + b*x + c*x^2)^2,x)
Output:
- 2*atanh((2*((d + e*x)^(1/2)*(36*a*c^2*e^6 - 18*b^2*c*e^6 - 36*c^3*d^2*e^ 4 + 36*b*c^2*d*e^5) + (9*(d + e*x)^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2 ) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*(-(9*(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2))) ^(1/2))/(54*c^2*d^2*e^6 + 54*a*c*e^8 - 54*b*c*d*e^7))*(-(9*(b^3*e^3 + e^3* (-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2))/( 8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2) - 2*atanh((2*((d + e*x)^(1/2) *(36*a*c^2*e^6 - 18*b^2*c*e^6 - 36*c^3*d^2*e^4 + 36*b*c^2*d*e^5) - (9*(d + e*x)^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4* d*e^2)*(e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e ^2 + 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*((9*(e^3*(-(4 *a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e ^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2))/(54*c^2*d^2*e^6 + 54*a *c*e^8 - 54*b*c*d*e^7))*((9*(e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a* b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2))/(8*(b^4*c + 16*a^2*c^3 - 8*a*b^2 *c^2)))^(1/2) - (e^2*(d + e*x)^(3/2))/((b*e - 2*c*d)*(d + e*x) + c*(d + e* x)^2 + a*e^2 + c*d^2 - b*d*e)
Time = 7.13 (sec) , antiderivative size = 4248, normalized size of antiderivative = 18.96 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x)
Output:
( - 12*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a* e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d *e + c*d**2) + b*e - 2*c*d))*a*c - 12*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c) *sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/s qrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b*c*x - 12*sqr t(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d *e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2* c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d** 2) + b*e - 2*c*d))*c**2*x**2 + 6*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d* e + c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d* *2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*a*b*e - 12*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a* e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b *d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sq rt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*a*c*d + 6*sqrt(c)*sqrt(2*sqrt( c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqrt( a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2* sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b**2*e*x - 12*sqr...