Integrand size = 22, antiderivative size = 322 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 e \sqrt {d+e x}}{c}-\frac {\sqrt {2} \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:
2*e*(e*x+d)^(1/2)/c-2^(1/2)*(2*c^2*d^2+b*(b-(-4*a*c+b^2)^(1/2))*e^2-2*c*e* (b*d-(-4*a*c+b^2)^(1/2)*d+a*e))*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))/c^(3/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-(b-( -4*a*c+b^2)^(1/2))*e)^(1/2)+2^(1/2)*(2*c^2*d^2+b*(b+(-4*a*c+b^2)^(1/2))*e^ 2-2*c*e*(b*d+(-4*a*c+b^2)^(1/2)*d+a*e))*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))/c^(3/2)/(-4*a*c+b^2)^(1/2)/(2* c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} e \sqrt {d+e x}+\frac {\left (-2 i c^2 d^2-b \left (i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (i b d+\sqrt {-b^2+4 a c} d+i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c^2 d^2-b \left (-i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (-i b d+\sqrt {-b^2+4 a c} d-i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{c^{3/2}} \] Input:
Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2),x]
Output:
(2*Sqrt[c]*e*Sqrt[d + e*x] + (((-2*I)*c^2*d^2 - b*(I*b + Sqrt[-b^2 + 4*a*c ])*e^2 + 2*c*e*(I*b*d + Sqrt[-b^2 + 4*a*c]*d + I*a*e))*ArcTan[(Sqrt[2]*Sqr t[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-1 /2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (((2*I)*c^2 *d^2 - b*((-I)*b + Sqrt[-b^2 + 4*a*c])*e^2 + 2*c*e*((-I)*b*d + Sqrt[-b^2 + 4*a*c]*d - I*a*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[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/c^(3/2)
Time = 0.59 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1146, 1197, 25, 27, 1480, 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.
\(\displaystyle \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx\) |
\(\Big \downarrow \) 1146 |
\(\displaystyle \frac {\int \frac {c d^2-a e^2+e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{c}+\frac {2 e \sqrt {d+e x}}{c}\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle \frac {2 \int -\frac {e \left (c d^2-b e d+a e^2-(2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}+\frac {2 e \sqrt {d+e x}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 e \sqrt {d+e x}}{c}-\frac {2 \int \frac {e \left (c d^2-b e d+a e^2-(2 c d-b e) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 e \sqrt {d+e x}}{c}-\frac {2 e \int \frac {c d^2-b e d+a e^2-(2 c d-b e) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle \frac {2 e \sqrt {d+e x}}{c}-\frac {2 e \left (-\frac {1}{2} \left (\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}-\frac {1}{2} \left (-\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 e \sqrt {d+e x}}{c}-\frac {2 e \left (\frac {\left (\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (-\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{e \sqrt {b^2-4 a c}}-b e+2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c}\) |
Input:
Int[(d + e*x)^(3/2)/(a + b*x + c*x^2),x]
Output:
(2*e*Sqrt[d + e*x])/c - (2*e*(((2*c*d - b*e + (2*c^2*d^2 + b^2*e^2 - 2*c*e *(b*d + a*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]) + ((2*c*d - b*e - (2*c^2*d^2 + b^2*e^2 - 2*c* e*(b*d + a*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])))/c
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_)^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[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol ] :> Simp[e*((d + e*x)^(m - 1)/(c*(m - 1))), x] + Simp[1/c Int[(d + e*x)^ (m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*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] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 1.23 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.07
method | result | size |
pseudoelliptic | \(\frac {e \left (2 \sqrt {e x +d}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\) | \(345\) |
derivativedivides | \(2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(353\) |
default | \(2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(353\) |
risch | \(\frac {2 e \sqrt {e x +d}}{c}-8 e \left (-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(354\) |
Input:
int((e*x+d)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
e/c*(2*(e*x+d)^(1/2)+(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2+(-e^2*(4*a*c-b ^2))^(1/2)*b*e-2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/(-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))+(2*a*c*e^2-b^ 2*e^2+2*b*c*d*e-2*c^2*d^2-(-e^2*(4*a*c-b^2))^(1/2)*b*e+2*(-e^2*(4*a*c-b^2) )^(1/2)*c*d)/(-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. 2770 vs. \(2 (272) = 544\).
Time = 0.25 (sec) , antiderivative size = 2770, normalized size of antiderivative = 8.60 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
-1/2*(sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^ 2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c ^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4 ))*log(sqrt(2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^2 - 3*(b^3*c - 4*a*b*c^2)*d*e^ 3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^4 - (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^ 3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2) *e^6)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2 *a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4* e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b* c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c ^3 - 4*a*c^4)) - 4*(3*c^3*d^4*e - 6*b*c^2*d^3*e^2 + 2*(2*b^2*c + a*c^2)*d^ 2*e^3 - (b^3 + 2*a*b*c)*d*e^4 + (a*b^2 - a^2*c)*e^5)*sqrt(e*x + d)) - sqrt (2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2* a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-sq rt(2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^2 - 3*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^4 - (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*...
\[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{a + b x + c x^{2}}\, dx \] Input:
integrate((e*x+d)**(3/2)/(c*x**2+b*x+a),x)
Output:
Integral((d + e*x)**(3/2)/(a + b*x + c*x**2), x)
\[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} + b x + a} \,d x } \] Input:
integrate((e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (272) = 544\).
Time = 0.41 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
2*sqrt(e*x + d)*e/c + 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e )*(2*(b^2*c - 4*a*c^2)*d*e - (b^3 - 4*a*b*c)*e^2)*c^2*e^2 - 2*(sqrt(b^2 - 4*a*c)*c^3*d^2*e - sqrt(b^2 - 4*a*c)*b*c^2*d*e^2 + sqrt(b^2 - 4*a*c)*a*c^2 *e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) - (4* c^5*d^3*e - 6*b*c^4*d^2*e^2 + 4*(b^2*c^3 - a*c^4)*d*e^3 - (b^3*c^2 - 2*a*b *c^3)*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^2*d - b*c*e + sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2))/((sqrt(b^2 - 4*a*c)*c^4*d^2 - s qrt(b^2 - 4*a*c)*b*c^3*d*e + sqrt(b^2 - 4*a*c)*a*c^3*e^2)*c^2*abs(e)) - 1/ 4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^2*c - 4*a*c^2)*d *e - (b^3 - 4*a*b*c)*e^2)*c^2*e^2 + 2*(sqrt(b^2 - 4*a*c)*c^3*d^2*e - sqrt( b^2 - 4*a*c)*b*c^2*d*e^2 + sqrt(b^2 - 4*a*c)*a*c^2*e^3)*sqrt(-4*c^2*d + 2* (b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) - (4*c^5*d^3*e - 6*b*c^4*d^2* e^2 + 4*(b^2*c^3 - a*c^4)*d*e^3 - (b^3*c^2 - 2*a*b*c^3)*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^2*d - b*c*e - sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2))/((sqrt(b^2 - 4*a*c)*c^4*d^2 - sqrt(b^2 - 4*a*c)*b*c^3*d *e + sqrt(b^2 - 4*a*c)*a*c^3*e^2)*c^2*abs(e))
Time = 6.95 (sec) , antiderivative size = 8334, normalized size of antiderivative = 25.88 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
int((d + e*x)^(3/2)/(a + b*x + c*x^2),x)
Output:
(2*e*(d + e*x)^(1/2))/c - atan(((((8*(4*a^2*c^3*e^5 - a*b^2*c^2*e^5 + 4*a* c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*b*c^3*d*e^4))/c - (8*( d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^ 4)))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2) )/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3) ^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2 *e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^( 1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^( 1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (8*(d + e*x)^(1/2)*(b^4*e^6 + 2*a^2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3* d^2*e^4 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d* e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2 *e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^ 3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e ^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e ^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5 + b^4*...
Time = 0.43 (sec) , antiderivative size = 2819, normalized size of antiderivative = 8.75 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^(3/2)/(c*x^2+b*x+a),x)
Output:
(2*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))*b*c*e - 4*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)*sq rt(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*d + 4*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*c*e* *2 - 2*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*sqr t(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**2 + 4*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*c*d*e - 4*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))*c**2*d**2 - 2*sqrt(2*sqrt(c)*sqrt(a*e*...