Integrand size = 40, antiderivative size = 434 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 c^{9/2} d^{9/2} e^{11/2}} \] Output:
1/512*(-7*a^4*e^8-8*a^3*c*d^2*e^6-6*a^2*c^2*d^4*e^4+21*c^4*d^8)*(2*c*d*e*x +a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^5+1/20*(a/ c/d-3*d/e^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/6*x^3*(a*d*e+(a *e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e-1/960*(105*c^3*d^6-21*a*c^2*d^4*e^2-33*a^ 2*c*d^2*e^4-35*a^3*e^6-6*c*d*e*(-7*a^2*e^4-6*a*c*d^2*e^2+21*c^2*d^4)*x)*(a *d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/e^4-1/512*(-a*e^2+c*d^2)^3*( 7*a^3*e^6+15*a^2*c*d^2*e^4+21*a*c^2*d^4*e^2+21*c^3*d^6)*arctanh(c^(1/2)*d^ (1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^( 9/2)/e^(11/2)
Time = 2.05 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10}+5 a^4 c d e^8 (11 d+14 e x)+2 a^3 c^2 d^2 e^6 \left (27 d^2-16 d e x-28 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (13 d^3-6 d^2 e x+4 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (-525 d^4+336 d^3 e x-264 d^2 e^2 x^2+224 d e^3 x^3+1664 e^4 x^4\right )+c^5 d^5 \left (315 d^5-210 d^4 e x+168 d^3 e^2 x^2-144 d^2 e^3 x^3+128 d e^4 x^4+1280 e^5 x^5\right )\right )-\frac {15 \left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 c^{9/2} d^{9/2} e^{11/2}} \] Input:
Integrate[(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x),x]
Output:
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10 + 5 *a^4*c*d*e^8*(11*d + 14*e*x) + 2*a^3*c^2*d^2*e^6*(27*d^2 - 16*d*e*x - 28*e ^2*x^2) + 6*a^2*c^3*d^3*e^4*(13*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 + 8*e^3*x^3) + a*c^4*d^4*e^2*(-525*d^4 + 336*d^3*e*x - 264*d^2*e^2*x^2 + 224*d*e^3*x^3 + 1664*e^4*x^4) + c^5*d^5*(315*d^5 - 210*d^4*e*x + 168*d^3*e^2*x^2 - 144* d^2*e^3*x^3 + 128*d*e^4*x^4 + 1280*e^5*x^5)) - (15*(c*d^2 - a*e^2)^3*(21*c ^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e^6)*ArcTanh[(Sqrt[c] *Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*S qrt[d + e*x])))/(7680*c^(9/2)*d^(9/2)*e^(11/2))
Time = 1.29 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1215, 1236, 27, 1236, 27, 1225, 1087, 1092, 219}
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.
\(\displaystyle \int \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{d+e x} \, dx\) |
\(\Big \downarrow \) 1215 |
\(\displaystyle \int x^3 (a e+c d x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}dx\) |
\(\Big \downarrow \) 1236 |
\(\displaystyle \frac {\int -\frac {3}{2} c d x^2 \left (2 a d e+\left (3 c d^2-a e^2\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{6 c d e}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\int x^2 \left (2 a d e+\left (3 c d^2-a e^2\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{4 e}\) |
\(\Big \downarrow \) 1236 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {\int -\frac {1}{2} x \left (4 a d e \left (3 c d^2-a e^2\right )+\left (21 c^2 d^4-6 a c e^2 d^2-7 a^2 e^4\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{5 c d e}+\frac {1}{5} x^2 \left (\frac {3 d}{e}-\frac {a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {1}{5} x^2 \left (\frac {3 d}{e}-\frac {a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}-\frac {\int x \left (4 a d e \left (3 c d^2-a e^2\right )+\left (21 c^2 d^4-6 a c e^2 d^2-7 a^2 e^4\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 c d e}}{4 e}\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {1}{5} x^2 \left (\frac {3 d}{e}-\frac {a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}-\frac {\frac {5 \left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c^2 d^2 e^2}-\frac {\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}}{4 e}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {1}{5} x^2 \left (\frac {3 d}{e}-\frac {a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}-\frac {\frac {5 \left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c^2 d^2 e^2}-\frac {\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}}{4 e}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {1}{5} x^2 \left (\frac {3 d}{e}-\frac {a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}-\frac {\frac {5 \left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c^2 d^2 e^2}-\frac {\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}}{4 e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}-\frac {\frac {1}{5} x^2 \left (\frac {3 d}{e}-\frac {a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}-\frac {\frac {5 \left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c^2 d^2 e^2}-\frac {\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}}{4 e}\) |
Input:
Int[(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x),x]
Output:
(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*e) - ((((3*d)/e - ( a*e)/(c*d))*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/5 - (-1/24* ((105*c^3*d^6 - 21*a*c^2*d^4*e^2 - 33*a^2*c*d^2*e^4 - 35*a^3*e^6 - 6*c*d*e *(21*c^2*d^4 - 6*a*c*d^2*e^2 - 7*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(c^2*d^2*e^2) + (5*(21*c^4*d^8 - 6*a^2*c^2*d^4*e^4 - 8*a ^3*c*d^2*e^6 - 7*a^4*e^8)*(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^ 2 + a*e^2)*x + c*d*e*x^2])/(4*c*d*e) - ((c*d^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2 )*x + c*d*e*x^2])])/(8*c^(3/2)*d^(3/2)*e^(3/2))))/(16*c^2*d^2*e^2))/(10*c* d*e))/(4*e)
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( (d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
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)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1425\) vs. \(2(402)=804\).
Time = 2.58 (sec) , antiderivative size = 1426, normalized size of antiderivative = 3.29
Input:
int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d),x,method=_RETURNVE RBOSE)
Output:
d^2/e^3*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/ 2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+ c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a *e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e +(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))+1/e*(1/6*x*(a*d*e+(a*e^ 2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-7/12*(a*e^2+c*d^2)/d/e/c*(1/5*(a*d*e+(a* e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-1/2*(a*e^2+c*d^2)/d/e/c*(1/8*(2*c*d*e* x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d ^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c *d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*l n((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x ^2*e)^(1/2))/(d*e*c)^(1/2))))-1/6*a/c*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+ (a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2 )/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/ 2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2 +c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^( 1/2))))-d/e^2*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-1/2*(a*e^ 2+c*d^2)/d/e/c*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2 *e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x +a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*...
Time = 0.14 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.41 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:
integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),x, algorithm ="fricas")
Output:
[-1/30720*(15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 4*a^ 3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 7*a^6*e^12)*sqrt(c* d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(c* d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d* e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11 *e - 525*a*c^5*d^9*e^3 + 78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*a^4* c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 - 16*(9*c^6*d^8*e^4 - 14*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*x^3 + 8*(21*c^ 6*d^9*e^3 - 33*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9)*x^2 - 2*(105*c^6*d^10*e^2 - 168*a*c^5*d^8*e^4 + 18*a^2*c^4*d^6*e^6 + 16*a^3*c^ 3*d^4*e^8 - 35*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^ 2)*x))/(c^5*d^5*e^6), 1/15360*(15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^ 2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 7*a^6*e^12)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e ^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2 *e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^1 1*e - 525*a*c^5*d^9*e^3 + 78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*a^4 *c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 - 16*(9*c^6*d^8*e^4 - 14*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*x^3 + 8*(21*c ^6*d^9*e^3 - 33*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9)*...
Leaf count of result is larger than twice the leaf count of optimal. 984 vs. \(2 (442) = 884\).
Time = 82.71 (sec) , antiderivative size = 2502, normalized size of antiderivative = 5.76 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:
integrate(x**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d),x)
Output:
a*e*Piecewise(((-a*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c *d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c *d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c* d**2/2)*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a *d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c *d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/ (2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c *d**2))*(x**4/5 + x**3*(a*e**2/10 + c*d**2/10)/(4*c*d*e) + x**2*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e) + x*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e)) /(2*c*d*e) + (-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d* *2/2)/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(a*e**2/10 + c*d* *2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10 )*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(c*d*e)), Ne( c*d*e, 0)), (2*(-a**3*d**3*e**3*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 ...
Exception generated. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),x, algorithm ="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.15 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c d x + \frac {c^{6} d^{7} e^{4} + 13 \, a c^{5} d^{5} e^{6}}{c^{5} d^{5} e^{5}}\right )} x - \frac {9 \, c^{6} d^{8} e^{3} - 14 \, a c^{5} d^{6} e^{5} - 3 \, a^{2} c^{4} d^{4} e^{7}}{c^{5} d^{5} e^{5}}\right )} x + \frac {21 \, c^{6} d^{9} e^{2} - 33 \, a c^{5} d^{7} e^{4} + 3 \, a^{2} c^{4} d^{5} e^{6} - 7 \, a^{3} c^{3} d^{3} e^{8}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{6} d^{10} e - 168 \, a c^{5} d^{8} e^{3} + 18 \, a^{2} c^{4} d^{6} e^{5} + 16 \, a^{3} c^{3} d^{4} e^{7} - 35 \, a^{4} c^{2} d^{2} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {315 \, c^{6} d^{11} - 525 \, a c^{5} d^{9} e^{2} + 78 \, a^{2} c^{4} d^{7} e^{4} + 54 \, a^{3} c^{3} d^{5} e^{6} + 55 \, a^{4} c^{2} d^{3} e^{8} - 105 \, a^{5} c d e^{10}}{c^{5} d^{5} e^{5}}\right )} + \frac {{\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{4} d^{4} e^{5}} \] Input:
integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),x, algorithm ="giac")
Output:
1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*c*d*x + (c^6*d^7*e^4 + 13*a*c^5*d^5*e^6)/(c^5*d^5*e^5))*x - (9*c^6*d^8*e^3 - 14*a *c^5*d^6*e^5 - 3*a^2*c^4*d^4*e^7)/(c^5*d^5*e^5))*x + (21*c^6*d^9*e^2 - 33* a*c^5*d^7*e^4 + 3*a^2*c^4*d^5*e^6 - 7*a^3*c^3*d^3*e^8)/(c^5*d^5*e^5))*x - (105*c^6*d^10*e - 168*a*c^5*d^8*e^3 + 18*a^2*c^4*d^6*e^5 + 16*a^3*c^3*d^4* e^7 - 35*a^4*c^2*d^2*e^9)/(c^5*d^5*e^5))*x + (315*c^6*d^11 - 525*a*c^5*d^9 *e^2 + 78*a^2*c^4*d^7*e^4 + 54*a^3*c^3*d^5*e^6 + 55*a^4*c^2*d^3*e^8 - 105* a^5*c*d*e^10)/(c^5*d^5*e^5)) + 1/1024*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 1 5*a^2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^ 10 - 7*a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - s qrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^4*d^4*e^5)
Timed out. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {x^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \] Input:
int((x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x),x)
Output:
int((x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x), x)
Time = 0.35 (sec) , antiderivative size = 1033, normalized size of antiderivative = 2.38 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:
int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),x)
Output:
( - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d*e**11 + 55*sqrt(d + e*x)* sqrt(a*e + c*d*x)*a**4*c**2*d**3*e**9 + 70*sqrt(d + e*x)*sqrt(a*e + c*d*x) *a**4*c**2*d**2*e**10*x + 54*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d** 5*e**7 - 32*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**4*e**8*x - 56*sqr t(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**9*x**2 + 78*sqrt(d + e*x)*s qrt(a*e + c*d*x)*a**2*c**4*d**7*e**5 - 36*sqrt(d + e*x)*sqrt(a*e + c*d*x)* a**2*c**4*d**6*e**6*x + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**5* e**7*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**8*x**3 - 525*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**9*e**3 + 336*sqrt(d + e*x)*s qrt(a*e + c*d*x)*a*c**5*d**8*e**4*x - 264*sqrt(d + e*x)*sqrt(a*e + c*d*x)* a*c**5*d**7*e**5*x**2 + 224*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**6*e* *6*x**3 + 1664*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**5*e**7*x**4 + 315 *sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**11*e - 210*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**10*e**2*x + 168*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d** 9*e**3*x**2 - 144*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**8*e**4*x**3 + 12 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**7*e**5*x**4 + 1280*sqrt(d + e*x) *sqrt(a*e + c*d*x)*c**6*d**6*e**6*x**5 + 105*sqrt(e)*sqrt(d)*sqrt(c)*log(( sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c *d**2))*a**6*e**12 - 90*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c* d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**5*c*d**...