Integrand size = 25, antiderivative size = 581 \[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}-\frac {2 (c d+b e) (d+e x)^{5/2}}{5 c^2 e^2}+\frac {2 (d+e x)^{7/2}}{7 c e^2}+\frac {\sqrt {2} \left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )-\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 b^3 c d e-4 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-3 a e^2\right )+a c^2 \left (c d^2-a e^2\right )+\frac {2 b^4 c d e-8 a b^2 c^2 d e+4 a^2 c^3 d e-b^5 e^2-b^3 c \left (c d^2-5 a e^2\right )+a b c^2 \left (3 c d^2-5 a e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{9/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:
2*(2*a*b*c*e-a*c^2*d-b^3*e+b^2*c*d)*(e*x+d)^(1/2)/c^4+2/3*(-a*c+b^2)*(e*x+ d)^(3/2)/c^3-2/5*(b*e+c*d)*(e*x+d)^(5/2)/c^2/e^2+2/7*(e*x+d)^(7/2)/c/e^2+2 ^(1/2)*(2*b^3*c*d*e-4*a*b*c^2*d*e-b^4*e^2-b^2*c*(-3*a*e^2+c*d^2)+a*c^2*(-a *e^2+c*d^2)-(2*b^4*c*d*e-8*a*b^2*c^2*d*e+4*a^2*c^3*d*e-b^5*e^2-b^3*c*(-5*a *e^2+c*d^2)+a*b*c^2*(-5*a*e^2+3*c*d^2))/(-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^(9/2)/(2 *c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)+2^(1/2)*(2*b^3*c*d*e-4*a*b*c^2*d*e-b^ 4*e^2-b^2*c*(-3*a*e^2+c*d^2)+a*c^2*(-a*e^2+c*d^2)+(2*b^4*c*d*e-8*a*b^2*c^2 *d*e+4*a^2*c^3*d*e-b^5*e^2-b^3*c*(-5*a*e^2+c*d^2)+a*b*c^2*(-5*a*e^2+3*c*d^ 2))/(-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^(9/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
Result contains complex when optimal does not.
Time = 4.48 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.30 \[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=-\frac {2 \sqrt {d+e x} \left (105 b^3 e^3+3 c^3 (2 d-5 e x) (d+e x)^2-35 b c e^2 (4 b d+6 a e+b e x)+7 c^2 e \left (3 b (d+e x)^2+5 a e (4 d+e x)\right )\right )}{105 c^4 e^2}+\frac {\left (i b^5 e^2+b^4 e \left (-2 i c d+\sqrt {-b^2+4 a c} e\right )+i b^3 c \left (c d^2+e \left (2 i \sqrt {-b^2+4 a c} d-5 a e\right )\right )+a c^2 \left (a \sqrt {-b^2+4 a c} e^2-c d \left (\sqrt {-b^2+4 a c} d+4 i a e\right )\right )+a b c^2 \left (-3 i c d^2+e \left (4 \sqrt {-b^2+4 a c} d+5 i a e\right )\right )+b^2 c \left (-3 a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d+8 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (-i b^5 e^2+b^4 e \left (2 i c d+\sqrt {-b^2+4 a c} e\right )+a c^2 \left (a \sqrt {-b^2+4 a c} e^2+c d \left (-\sqrt {-b^2+4 a c} d+4 i a e\right )\right )+a b c^2 \left (3 i c d^2+e \left (4 \sqrt {-b^2+4 a c} d-5 i a e\right )\right )+b^3 c \left (-i c d^2+e \left (-2 \sqrt {-b^2+4 a c} d+5 i a e\right )\right )+b^2 c \left (-3 a \sqrt {-b^2+4 a c} e^2+c d \left (\sqrt {-b^2+4 a c} d-8 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{c^{9/2} \sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}} \] Input:
Integrate[(x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]
Output:
(-2*Sqrt[d + e*x]*(105*b^3*e^3 + 3*c^3*(2*d - 5*e*x)*(d + e*x)^2 - 35*b*c* e^2*(4*b*d + 6*a*e + b*e*x) + 7*c^2*e*(3*b*(d + e*x)^2 + 5*a*e*(4*d + e*x) )))/(105*c^4*e^2) + ((I*b^5*e^2 + b^4*e*((-2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e ) + I*b^3*c*(c*d^2 + e*((2*I)*Sqrt[-b^2 + 4*a*c]*d - 5*a*e)) + a*c^2*(a*Sq rt[-b^2 + 4*a*c]*e^2 - c*d*(Sqrt[-b^2 + 4*a*c]*d + (4*I)*a*e)) + a*b*c^2*( (-3*I)*c*d^2 + e*(4*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e)) + b^2*c*(-3*a*Sqrt[ -b^2 + 4*a*c]*e^2 + c*d*(Sqrt[-b^2 + 4*a*c]*d + (8*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]])/(c ^(9/2)*Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (((-I)*b^5*e^2 + b^4*e*((2*I)*c*d + Sqrt[-b^2 + 4*a*c]*e) + a*c^2*(a*Sq rt[-b^2 + 4*a*c]*e^2 + c*d*(-(Sqrt[-b^2 + 4*a*c]*d) + (4*I)*a*e)) + a*b*c^ 2*((3*I)*c*d^2 + e*(4*Sqrt[-b^2 + 4*a*c]*d - (5*I)*a*e)) + b^3*c*((-I)*c*d ^2 + e*(-2*Sqrt[-b^2 + 4*a*c]*d + (5*I)*a*e)) + b^2*c*(-3*a*Sqrt[-b^2 + 4* a*c]*e^2 + c*d*(Sqrt[-b^2 + 4*a*c]*d - (8*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]])/(c^(9/2)*Sq rt[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e])
Time = 8.06 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {2 \int \left (\frac {(d+e x)^3}{c e}-\frac {(c d+b e) (d+e x)^2}{c^2 e}+\frac {\left (b^2-a c\right ) e (d+e x)}{c^3}+\frac {e \left (-e b^3+c d b^2+2 a c e b-a c^2 d\right )}{c^4}-\frac {\left (-e b^3+c d b^2+2 a c e b-a c^2 d\right ) \left (c d^2-b e d+a e^2\right )+\left (-e^2 b^4+2 c d e b^3-c \left (c d^2-3 a e^2\right ) b^2-4 a c^2 d e b+a c^2 \left (c d^2-a e^2\right )\right ) (d+e x)}{c^4 e \left (\frac {c (d+e x)^2}{e^2}-\frac {(2 c d-b e) (d+e x)}{e^2}+a+\frac {d (c d-b e)}{e^2}\right )}\right )d\sqrt {d+e x}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {e \left (-\frac {4 a^2 c^3 d e-b^3 c \left (c d^2-5 a e^2\right )-8 a b^2 c^2 d e+a b c^2 \left (3 c d^2-5 a e^2\right )+b^5 \left (-e^2\right )+2 b^4 c d e}{\sqrt {b^2-4 a c}}-b^2 c \left (c d^2-3 a e^2\right )-4 a b c^2 d e+a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{9/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {e \left (\frac {4 a^2 c^3 d e-b^3 c \left (c d^2-5 a e^2\right )-8 a b^2 c^2 d e+a b c^2 \left (3 c d^2-5 a e^2\right )+b^5 \left (-e^2\right )+2 b^4 c d e}{\sqrt {b^2-4 a c}}-b^2 c \left (c d^2-3 a e^2\right )-4 a b c^2 d e+a c^2 \left (c d^2-a e^2\right )+b^4 \left (-e^2\right )+2 b^3 c d e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{9/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {e \left (b^2-a c\right ) (d+e x)^{3/2}}{3 c^3}+\frac {e \sqrt {d+e x} \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}-\frac {(d+e x)^{5/2} (b e+c d)}{5 c^2 e}+\frac {(d+e x)^{7/2}}{7 c e}\right )}{e}\) |
Input:
Int[(x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]
Output:
(2*((e*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*Sqrt[d + e*x])/c^4 + ((b^2 - a*c)*e*(d + e*x)^(3/2))/(3*c^3) - ((c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e ) + (d + e*x)^(7/2)/(7*c*e) + (e*(2*b^3*c*d*e - 4*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 3*a*e^2) + a*c^2*(c*d^2 - a*e^2) - (2*b^4*c*d*e - 8*a*b^2*c ^2*d*e + 4*a^2*c^3*d*e - b^5*e^2 - b^3*c*(c*d^2 - 5*a*e^2) + a*b*c^2*(3*c* d^2 - 5*a*e^2))/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^(9/2)*Sqrt[2*c*d - ( b - Sqrt[b^2 - 4*a*c])*e]) + (e*(2*b^3*c*d*e - 4*a*b*c^2*d*e - b^4*e^2 - b ^2*c*(c*d^2 - 3*a*e^2) + a*c^2*(c*d^2 - a*e^2) + (2*b^4*c*d*e - 8*a*b^2*c^ 2*d*e + 4*a^2*c^3*d*e - b^5*e^2 - b^3*c*(c*d^2 - 5*a*e^2) + a*b*c^2*(3*c*d ^2 - 5*a*e^2))/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^(9/2)*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 = 1.87 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(\frac {-e^{2} \left (\left (\left (a^{2} c^{2}-3 c a \,b^{2}+b^{4}\right ) e^{2}+2 d \left (2 a b \,c^{2}-b^{3} c \right ) e -c^{2} d^{2} \left (a c -b^{2}\right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-5 e \left (b \left (a^{2} c^{2}-c a \,b^{2}+\frac {1}{5} b^{4}\right ) e^{2}-\frac {4 d \left (a^{2} c^{2}-2 c a \,b^{2}+\frac {1}{2} b^{4}\right ) c e}{5}-\frac {3 d^{2} b \left (a c -\frac {b^{2}}{3}\right ) c^{2}}{5}\right )\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \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 )+\left (e^{2} \left (\left (\left (a^{2} c^{2}-3 c a \,b^{2}+b^{4}\right ) e^{2}+2 d \left (2 a b \,c^{2}-b^{3} c \right ) e -c^{2} d^{2} \left (a c -b^{2}\right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+5 e \left (b \left (a^{2} c^{2}-c a \,b^{2}+\frac {1}{5} b^{4}\right ) e^{2}-\frac {4 d \left (a^{2} c^{2}-2 c a \,b^{2}+\frac {1}{2} b^{4}\right ) c e}{5}-\frac {3 d^{2} b \left (a c -\frac {b^{2}}{3}\right ) c^{2}}{5}\right )\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 )+4 \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\frac {c^{3} x^{3}}{14}-\frac {x \left (\frac {3 b x}{5}+a \right ) c^{2}}{6}+b \left (\frac {b x}{6}+a \right ) c -\frac {b^{3}}{2}\right ) e^{3}-\frac {2 d c \left (-\frac {6 c^{2} x^{2}}{35}+\left (\frac {3 b x}{10}+a \right ) c -b^{2}\right ) e^{2}}{3}-\frac {d^{2} \left (-\frac {c x}{7}+b \right ) c^{2} e}{10}-\frac {d^{3} c^{3}}{35}\right ) \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}}{\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 )}\, c^{4} e^{2}}\) | \(680\) |
| risch | \(\frac {2 \left (15 c^{3} x^{3} e^{3}-21 b \,c^{2} e^{3} x^{2}+24 c^{3} d \,e^{2} x^{2}-35 a \,c^{2} e^{3} x +35 x \,b^{2} c \,e^{3}-42 b \,c^{2} d \,e^{2} x +3 d^{2} e \,c^{3} x +210 a b c \,e^{3}-140 d \,e^{2} a \,c^{2}-105 b^{3} e^{3}+140 d \,e^{2} b^{2} c -21 d^{2} e b \,c^{2}-6 d^{3} c^{3}\right ) \sqrt {e x +d}}{105 e^{2} c^{4}}+\frac {-\frac {\left (-5 a^{2} b \,c^{2} e^{3}+4 a^{2} c^{3} d \,e^{2}+5 a \,b^{3} c \,e^{3}-8 a \,b^{2} c^{2} d \,e^{2}+3 a b \,d^{2} e \,c^{3}-b^{5} e^{3}+2 b^{4} c d \,e^{2}-b^{3} c^{2} d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}+4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\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 )}{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 (5 a^{2} b \,c^{2} e^{3}-4 a^{2} c^{3} d \,e^{2}-5 a \,b^{3} c \,e^{3}+8 a \,b^{2} c^{2} d \,e^{2}-3 a b \,d^{2} e \,c^{3}+b^{5} e^{3}-2 b^{4} c d \,e^{2}+b^{3} c^{2} d^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}+4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\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 )}{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}}}{c^{3}}\) | \(888\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-a \,c^{2} d \,e^{2} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}+b^{2} c d \,e^{2} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{2} \left (-\frac {\left (5 a^{2} b \,c^{2} e^{3}-4 a^{2} c^{3} d \,e^{2}-5 a \,b^{3} c \,e^{3}+8 a \,b^{2} c^{2} d \,e^{2}-3 a b \,d^{2} e \,c^{3}+b^{5} e^{3}-2 b^{4} c d \,e^{2}+b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\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 (-5 a^{2} b \,c^{2} e^{3}+4 a^{2} c^{3} d \,e^{2}+5 a \,b^{3} c \,e^{3}-8 a \,b^{2} c^{2} d \,e^{2}+3 a b \,d^{2} e \,c^{3}-b^{5} e^{3}+2 b^{4} c d \,e^{2}-b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\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^{3}}}{e^{2}}\) | \(893\) |
| default | \(\frac {\frac {2 \left (\frac {\left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}-\frac {b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a b c \,e^{3} \sqrt {e x +d}-a \,c^{2} d \,e^{2} \sqrt {e x +d}-b^{3} e^{3} \sqrt {e x +d}+b^{2} c d \,e^{2} \sqrt {e x +d}\right )}{c^{4}}-\frac {8 e^{2} \left (-\frac {\left (5 a^{2} b \,c^{2} e^{3}-4 a^{2} c^{3} d \,e^{2}-5 a \,b^{3} c \,e^{3}+8 a \,b^{2} c^{2} d \,e^{2}-3 a b \,d^{2} e \,c^{3}+b^{5} e^{3}-2 b^{4} c d \,e^{2}+b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\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 (-5 a^{2} b \,c^{2} e^{3}+4 a^{2} c^{3} d \,e^{2}+5 a \,b^{3} c \,e^{3}-8 a \,b^{2} c^{2} d \,e^{2}+3 a b \,d^{2} e \,c^{3}-b^{5} e^{3}+2 b^{4} c d \,e^{2}-b^{3} c^{2} d^{2} e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a^{2} c^{2} e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,b^{2} c \,e^{2}-4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b \,c^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{3} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{4} e^{2}+2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c^{2} d^{2}\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^{3}}}{e^{2}}\) | \(893\) |
Input:
int(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
(-e^2*(((a^2*c^2-3*a*b^2*c+b^4)*e^2+2*d*(2*a*b*c^2-b^3*c)*e-c^2*d^2*(a*c-b ^2))*(-4*e^2*(a*c-1/4*b^2))^(1/2)-5*e*(b*(a^2*c^2-c*a*b^2+1/5*b^4)*e^2-4/5 *d*(a^2*c^2-2*c*a*b^2+1/2*b^4)*c*e-3/5*d^2*b*(a*c-1/3*b^2)*c^2))*((b*e-2*c *d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*2^(1/2)*arctanh((e*x+d)^(1/2)*c* 2^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+(e^2*(((a^2*c ^2-3*a*b^2*c+b^4)*e^2+2*d*(2*a*b*c^2-b^3*c)*e-c^2*d^2*(a*c-b^2))*(-4*e^2*( a*c-1/4*b^2))^(1/2)+5*e*(b*(a^2*c^2-c*a*b^2+1/5*b^4)*e^2-4/5*d*(a^2*c^2-2* c*a*b^2+1/2*b^4)*c*e-3/5*d^2*b*(a*c-1/3*b^2)*c^2))*2^(1/2)*arctan((e*x+d)^ (1/2)*c*2^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+4*(e*x +d)^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((1/14*c^3*x^ 3-1/6*x*(3/5*b*x+a)*c^2+b*(1/6*b*x+a)*c-1/2*b^3)*e^3-2/3*d*c*(-6/35*c^2*x^ 2+(3/10*b*x+a)*c-b^2)*e^2-1/10*d^2*(-1/7*c*x+b)*c^2*e-1/35*d^3*c^3)*(-4*e^ 2*(a*c-1/4*b^2))^(1/2))*((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2 ))/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/((-b*e+2*c*d+(-4*e^2 *(a*c-1/4*b^2))^(1/2))*c)^(1/2)/(-4*e^2*(a*c-1/4*b^2))^(1/2)/c^4/e^2
Leaf count of result is larger than twice the leaf count of optimal. 11459 vs. \(2 (527) = 1054\).
Time = 6.58 (sec) , antiderivative size = 11459, normalized size of antiderivative = 19.72 \[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Timed out} \] Input:
integrate(x**3*(e*x+d)**(3/2)/(c*x**2+b*x+a),x)
Output:
Timed out
\[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} x^{3}}{c x^{2} + b x + a} \,d x } \] Input:
integrate(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
integrate((e*x + d)^(3/2)*x^3/(c*x^2 + b*x + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 1397 vs. \(2 (527) = 1054\).
Time = 0.35 (sec) , antiderivative size = 1397, normalized size of antiderivative = 2.40 \[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(e*x+d)^(3/2)/(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^4*c^2 - 5*a*b^2* c^3 + 4*a^2*c^4)*d^2 - 2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^2)*c^2*e^2 - 2*((b^2*c^4 - a*c^5 )*sqrt(b^2 - 4*a*c)*d^3 - (2*b^3*c^3 - 3*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2*e + (b^4*c^2 - a*b^2*c^3 - a^2*c^4)*sqrt(b^2 - 4*a*c)*d*e^2 - (a*b^3*c^2 - 2 *a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a *c)*c)*e)*abs(c)*abs(e) + (2*(b^3*c^5 - 3*a*b*c^6)*d^3*e - (5*b^4*c^4 - 19 *a*b^2*c^5 + 8*a^2*c^6)*d^2*e^2 + 2*(2*b^5*c^3 - 9*a*b^3*c^4 + 7*a^2*b*c^5 )*d*e^3 - (b^6*c^2 - 5*a*b^4*c^3 + 5*a^2*b^2*c^4)*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 ^8*d*e^16 - b*c^7*e^17 + sqrt(-4*(c^8*d^2*e^16 - b*c^7*d*e^17 + a*c^7*e^18 )*c^8*e^16 + (2*c^8*d*e^16 - b*c^7*e^17)^2))/(c^8*e^16)))/((sqrt(b^2 - 4*a *c)*c^7*d^2 - sqrt(b^2 - 4*a*c)*b*c^6*d*e + sqrt(b^2 - 4*a*c)*a*c^6*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^4*c ^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2 - 2*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)* d*e + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^2)*c^2*e^2 + 2*((b^ 2*c^4 - a*c^5)*sqrt(b^2 - 4*a*c)*d^3 - (2*b^3*c^3 - 3*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^2*e + (b^4*c^2 - a*b^2*c^3 - a^2*c^4)*sqrt(b^2 - 4*a*c)*d*e^2 - ( a*b^3*c^2 - 2*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + s qrt(b^2 - 4*a*c)*c)*e)*abs(c)*abs(e) + (2*(b^3*c^5 - 3*a*b*c^6)*d^3*e -...
Time = 15.06 (sec) , antiderivative size = 25497, normalized size of antiderivative = 43.88 \[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \] Input:
int((x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x)
Output:
atan(((((8*(4*a^3*c^8*d*e^4 - 8*a^3*b*c^7*e^5 - a*b^5*c^5*e^5 + b^6*c^5*d* e^4 + 6*a^2*b^3*c^6*e^5 + 4*a^2*c^9*d^3*e^2 + b^4*c^7*d^3*e^2 - 2*b^5*c^6* d^2*e^3 - 5*a*b^4*c^6*d*e^4 - 5*a*b^2*c^8*d^3*e^2 + 11*a*b^3*c^7*d^2*e^3 - 12*a^2*b*c^8*d^2*e^3 + 3*a^2*b^2*c^7*d*e^4))/c^7 - (8*(d + e*x)^(1/2)*(-( b^11*e^3 - 8*a^4*c^7*d^3 - b^8*c^3*d^3 + b^8*e^3*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^6*c^4*d^3 - 36*a^5*b*c^5*e^3 + 24*a^5*c^6*d*e^2 + 3*b^9*c^2*d^2*e - 33*a^2*b^4*c^5*d^3 + 38*a^3*b^2*c^6*d^3 + 63*a^2*b^7*c^2*e^3 - 138*a^3* b^5*c^3*e^3 + 129*a^4*b^3*c^4*e^3 + a^4*c^4*e^3*(-(4*a*c - b^2)^3)^(1/2) - b^5*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 13*a*b^9*c*e^3 - 3*b^10*c*d*e^2 + 15*a^2*b^4*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a^3*b^2*c^3*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^6*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 33*a*b^7*c^3*d^ 2*e + 36*a*b^8*c^2*d*e^2 + 84*a^4*b*c^6*d^2*e - 3*b^7*c*d*e^2*(-(4*a*c - b ^2)^3)^(1/2) + 4*a*b^3*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^5*d^3* (-(4*a*c - b^2)^3)^(1/2) + 126*a^2*b^5*c^4*d^2*e - 156*a^2*b^6*c^3*d*e^2 - 189*a^3*b^3*c^5*d^2*e + 288*a^3*b^4*c^4*d*e^2 - 198*a^4*b^2*c^5*d*e^2 - 3 *a^3*c^5*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 3*b^6*c^2*d^2*e*(-(4*a*c - b^2)^ 3)^(1/2) - 15*a*b^4*c^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^5*c^2*d*e^ 2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^3*b*c^4*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a^2*b^2*c^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 30*a^2*b^3*c^3*d*e^2*(-(4 *a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^11 + b^4*c^9 - 8*a*b^2*c^10)))^(1/2)...
\[ \int \frac {x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int \frac {x^{3} \left (e x +d \right )^{\frac {3}{2}}}{c \,x^{2}+b x +a}d x \] Input:
int(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x)
Output:
int(x^3*(e*x+d)^(3/2)/(c*x^2+b*x+a),x)