Integrand size = 25, antiderivative size = 316 \[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {2 b \sqrt {d+e x}}{c^2}+\frac {2 (d+e x)^{3/2}}{3 c e}+\frac {\sqrt {2} \left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:
-2*b*(e*x+d)^(1/2)/c^2+2/3*(e*x+d)^(3/2)/c/e+2^(1/2)*(b*c*d-b^2*e+a*c*e-(3 *a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(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))/c^(5/2)/(2*c*d-( b-(-4*a*c+b^2)^(1/2))*e)^(1/2)+2^(1/2)*(b*c*d-b^2*e+a*c*e+(3*a*b*c*e-2*a*c ^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(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))/c^(5/2)/(2*c*d-(b+(-4*a*c+b^2) ^(1/2))*e)^(1/2)
Time = 1.58 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.19 \[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x} (-3 b e+c (d+e x))}{e}+\frac {3 \sqrt {2} \left (-b^3 e+b c \left (-\sqrt {b^2-4 a c} d+3 a e\right )+b^2 \left (c d+\sqrt {b^2-4 a c} e\right )-a c \left (2 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {2} \left (b^3 e-b c \left (\sqrt {b^2-4 a c} d+3 a e\right )+a c \left (2 c d-\sqrt {b^2-4 a c} e\right )+b^2 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{3 c^{5/2}} \] Input:
Integrate[(x^2*Sqrt[d + e*x])/(a + b*x + c*x^2),x]
Output:
((2*Sqrt[c]*Sqrt[d + e*x]*(-3*b*e + c*(d + e*x)))/e + (3*Sqrt[2]*(-(b^3*e) + b*c*(-(Sqrt[b^2 - 4*a*c]*d) + 3*a*e) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*c*(2*c*d + 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[b^2 - 4*a*c]*Sqrt[-2*c* d + (b - Sqrt[b^2 - 4*a*c])*e]) + (3*Sqrt[2]*(b^3*e - b*c*(Sqrt[b^2 - 4*a* c]*d + 3*a*e) + a*c*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b^2*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sq rt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a *c])*e]))/(3*c^(5/2))
Time = 2.00 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1199, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {2 \int \left (-\frac {b e}{c^2}+\frac {d+e x}{c}+\frac {b \left (c d^2-b e d+a e^2\right )-\left (-e b^2+c d b+a c e\right ) (d+e x)}{c^2 \left (\frac {c (d+e x)^2}{e^2}-\frac {(2 c d-b e) (d+e x)}{e^2}+a+\frac {d (c d-b e)}{e^2}\right ) e}\right )d\sqrt {d+e x}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {e \left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {e \left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {b e \sqrt {d+e x}}{c^2}+\frac {(d+e x)^{3/2}}{3 c}\right )}{e}\) |
Input:
Int[(x^2*Sqrt[d + e*x])/(a + b*x + c*x^2),x]
Output:
(2*(-((b*e*Sqrt[d + e*x])/c^2) + (d + e*x)^(3/2)/(3*c) + (e*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*Ar cTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c]) *e]])/(Sqrt[2]*c^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (e*(b*c* d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4 *a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))) /e
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Time = 1.58 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(-\frac {2 \left (-c e x +3 b e -c d \right ) \sqrt {e x +d}}{3 e \,c^{2}}-\frac {8 \left (-\frac {\left (-3 a b c \,e^{2}+2 a \,c^{2} d e +e^{2} b^{3}-b^{2} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (3 a b c \,e^{2}-2 a \,c^{2} d e -e^{2} b^{3}+b^{2} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\) | \(425\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} c}{3}+\sqrt {e x +d}\, b e \right )}{c^{2}}+\frac {8 e \left (\frac {\left (-3 a b c \,e^{2}+2 a \,c^{2} d e +e^{2} b^{3}-b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{2}-2 a \,c^{2} d e -e^{2} b^{3}+b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}}{e}\) | \(430\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} c}{3}+\sqrt {e x +d}\, b e \right )}{c^{2}}+\frac {8 e \left (\frac {\left (-3 a b c \,e^{2}+2 a \,c^{2} d e +e^{2} b^{3}-b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (3 a b c \,e^{2}-2 a \,c^{2} d e -e^{2} b^{3}+b^{2} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}}{e}\) | \(430\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {e \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \left (\left (\left (a e +b d \right ) c -e \,b^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+2 a \,c^{2} d e +\left (-3 a b \,e^{2}-d e \,b^{2}\right ) c +e^{2} b^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{2}+\left (\frac {e \left (\left (\left (a e +b d \right ) c -e \,b^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-2 a \,c^{2} d e +\left (3 a b \,e^{2}+d e \,b^{2}\right ) c -e^{2} b^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )}{2}+\sqrt {e x +d}\, \left (\frac {\left (-e x -d \right ) c}{3}+b e \right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \,c^{2}}\) | \(436\) |
Input:
int(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-2/3*(-c*e*x+3*b*e-c*d)*(e*x+d)^(1/2)/e/c^2-8/c*(-1/8*(-3*a*b*c*e^2+2*a*c^ 2*d*e+e^2*b^3-b^2*c*d*e+(-e^2*(4*a*c-b^2))^(1/2)*a*c*e-(-e^2*(4*a*c-b^2))^ (1/2)*b^2*e+(-e^2*(4*a*c-b^2))^(1/2)*b*c*d)/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*(3*a*b*c*e ^2-2*a*c^2*d*e-e^2*b^3+b^2*c*d*e+(-e^2*(4*a*c-b^2))^(1/2)*a*c*e-(-e^2*(4*a *c-b^2))^(1/2)*b^2*e+(-e^2*(4*a*c-b^2))^(1/2)*b*c*d)/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. 2966 vs. \(2 (270) = 540\).
Time = 0.17 (sec) , antiderivative size = 2966, normalized size of antiderivative = 9.39 \[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
1/6*(3*sqrt(2)*c^2*e*sqrt(((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5* a*b^3*c + 5*a^2*b*c^2)*e + (b^2*c^5 - 4*a*c^6)*sqrt(((b^6*c^2 - 4*a*b^4*c^ 3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b* c^4)*d*e + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^ 2)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(sqrt(2)*((b^6*c - 6*a* b^4*c^2 + 8*a^2*b^2*c^3)*d - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c ^3)*e - (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c^7)*sqrt(((b^6*c^2 - 4*a*b^4*c^3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4 )*d*e + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^2)/ (b^2*c^10 - 4*a*c^11)))*sqrt(((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e + (b^2*c^5 - 4*a*c^6)*sqrt(((b^6*c^2 - 4*a*b^4 *c^3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3 *b*c^4)*d*e + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4) *e^2)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6)) - 4*((a^2*b^3*c - 2*a^3 *b*c^2)*d - (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e)*sqrt(e*x + d)) - 3*sqrt(2 )*c^2*e*sqrt(((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a ^2*b*c^2)*e + (b^2*c^5 - 4*a*c^6)*sqrt(((b^6*c^2 - 4*a*b^4*c^3 + 4*a^2*b^2 *c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e + (b ^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^2)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-sqrt(2)*((b^6*c - 6*a*b^4*c^2 +...
\[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{2} \sqrt {d + e x}}{a + b x + c x^{2}}\, dx \] Input:
integrate(x**2*(e*x+d)**(1/2)/(c*x**2+b*x+a),x)
Output:
Integral(x**2*sqrt(d + e*x)/(a + b*x + c*x**2), x)
\[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d} x^{2}}{c x^{2} + b x + a} \,d x } \] Input:
integrate(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)*x^2/(c*x^2 + b*x + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (270) = 540\).
Time = 0.30 (sec) , antiderivative size = 897, normalized size of antiderivative = 2.84 \[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
integrate(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
-1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^3*c - 4*a*b*c^2 )*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*c^2*e^2 - 2*(sqrt(b^2 - 4*a*c)*b*c^ 3*d^2 - sqrt(b^2 - 4*a*c)*b^2*c^2*d*e + sqrt(b^2 - 4*a*c)*a*b*c^2*e^2)*sqr t(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^2*c^4 - 2*a*c^5)*d^2*e - (3*b^3*c^3 - 8*a*b*c^4)*d*e^2 + (b^4*c^2 - 3*a*b^2*c^3) *e^3)*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^4*d*e^4 - b*c^3*e^5 + sqrt(-4*(c^4*d^2*e^4 - b*c ^3*d*e^5 + a*c^3*e^6)*c^4*e^4 + (2*c^4*d*e^4 - b*c^3*e^5)^2))/(c^4*e^4)))/ ((sqrt(b^2 - 4*a*c)*c^5*d^2 - sqrt(b^2 - 4*a*c)*b*c^4*d*e + sqrt(b^2 - 4*a *c)*a*c^4*e^2)*c^2*abs(e)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a* c)*c)*e)*((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*c^2*e^2 + 2*(sqrt(b^2 - 4*a*c)*b*c^3*d^2 - sqrt(b^2 - 4*a*c)*b^2*c^2*d*e + sqrt(b ^2 - 4*a*c)*a*b*c^2*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)* abs(c)*abs(e) + (2*(b^2*c^4 - 2*a*c^5)*d^2*e - (3*b^3*c^3 - 8*a*b*c^4)*d*e ^2 + (b^4*c^2 - 3*a*b^2*c^3)*e^3)*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^4*d*e^4 - b*c^3*e^5 - sqrt(-4*(c^4*d^2*e^4 - b*c^3*d*e^5 + a*c^3*e^6)*c^4*e^4 + (2*c^4*d*e^4 - b*c^3*e^5)^2))/(c^4*e^4)))/((sqrt(b^2 - 4*a*c)*c^5*d^2 - sqrt(b^2 - 4*a*c )*b*c^4*d*e + sqrt(b^2 - 4*a*c)*a*c^4*e^2)*c^2*abs(e)) + 2/3*((e*x + d)^(3 /2)*c^2*e^2 - 3*sqrt(e*x + d)*b*c*e^3)/(c^3*e^3)
Time = 11.87 (sec) , antiderivative size = 8171, normalized size of antiderivative = 25.86 \[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
int((x^2*(d + e*x)^(1/2))/(a + b*x + c*x^2),x)
Output:
(2*(d + e*x)^(3/2))/(3*c*e) - atan(((((8*(a*b^3*c^3*e^4 - 4*a^2*b*c^4*e^4 - b^4*c^3*d*e^3 + b^3*c^4*d^2*e^2 - 4*a*b*c^5*d^2*e^2 + 4*a*b^2*c^4*d*e^3) )/c^3 - (8*(d + e*x)^(1/2)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^ 3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4* a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c *d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b ^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6))) ^(1/2)*(b^3*c^5*e^3 - 2*b^2*c^6*d*e^2 - 4*a*b*c^6*e^3 + 8*a*c^7*d*e^2))/c^ 3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18* a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9* a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1 /2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3 )^(1/2))/(2*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2) - (8*(d + e*x)^(1 /2)*(b^6*e^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^2*e^2 + b^4 *c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e^3 - 10*a^2 *b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c^3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*( -(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3* b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3 )^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^7 + b^4*c^...
\[ \int \frac {x^2 \sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {x^{2} \sqrt {e x +d}}{c \,x^{2}+b x +a}d x \] Input:
int(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a),x)
Output:
int(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a),x)