Integrand size = 37, antiderivative size = 277 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {35 c^2 d^2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2}+\frac {35 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {14 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {35 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{4 e^{9/2}} \] Output:
35/4*c^2*d^2*(a-c*d^2/e^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^2+35/ 6*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^3/(e*x+d)-14/3*c*d*(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e^2/(e*x+d)^3-2/3*(a*d*e+(a*e^2+c*d^2 )*x+c*d*e*x^2)^(7/2)/e/(e*x+d)^5+35/4*c^(3/2)*d^(3/2)*(-a*e^2+c*d^2)^2*arc tanh(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/(e*x+ d))/e^(9/2)
Time = 0.48 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {e} \left (8 a^3 e^6+8 a^2 c d e^4 (7 d+10 e x)-a c^2 d^2 e^2 \left (175 d^2+238 d e x+39 e^2 x^2\right )+c^3 d^3 \left (105 d^3+140 d^2 e x+21 d e^2 x^2-6 e^3 x^3\right )\right )}{(d+e x)^2}+\frac {105 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{12 e^{9/2}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^6,x]
Output:
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(8*a^3*e^6 + 8*a^2*c*d*e^4*(7*d + 10*e*x) - a*c^2*d^2*e^2*(175*d^2 + 238*d*e*x + 39*e^2*x^2) + c^3*d^3*(1 05*d^3 + 140*d^2*e*x + 21*d*e^2*x^2 - 6*e^3*x^3)))/(d + e*x)^2) + (105*c^( 3/2)*d^(3/2)*(c*d^2 - a*e^2)^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sq rt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(12*e^(9/2) )
Time = 1.07 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1130, 1125, 25, 2192, 27, 1160, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1130 |
\(\displaystyle \frac {7 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^4}dx}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 1125 |
\(\displaystyle \frac {7 c d \left (-\frac {\int -\frac {c^3 d^3 x^2 e^5-c^2 d^2 \left (c d^2-3 a e^2\right ) x e^4+c d \left (c^2 d^4-3 a c e^2 d^2+3 a^2 e^4\right ) e^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {7 c d \left (\frac {\int \frac {c^3 d^3 x^2 e^5-c^2 d^2 \left (c d^2-3 a e^2\right ) x e^4+c d \left (c^2 d^4-3 a c e^2 d^2+3 a^2 e^4\right ) e^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {7 c d \left (\frac {\frac {\int \frac {c^2 d^2 e^4 \left (2 \left (2 c d^2-3 a e^2\right ) \left (c d^2-2 a e^2\right )-c d e \left (7 c d^2-9 a e^2\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \int \frac {2 \left (2 c d^2-3 a e^2\right ) \left (c d^2-2 a e^2\right )-c d e \left (7 c d^2-9 a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \left (\frac {15}{2} \left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\left (7 c d^2-9 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \left (15 \left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\left (7 c d^2-9 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \left (\frac {15 \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {e}}-\left (7 c d^2-9 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^6,x]
Output:
(-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3*e*(d + e*x)^5) + (7* c*d*((-2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e ^3*(d + e*x)) + ((c^2*d^2*e^4*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 ])/2 + (c*d*e^3*(-((7*c*d^2 - 9*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* d*e*x^2]) + (15*(c*d^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*S qrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*S qrt[c]*Sqrt[d]*Sqrt[e])))/4)/e^6))/(3*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[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[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2) Int[(1/Sqrt[a + b*x + c*x^2])*Expan dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x ), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && EqQ[m + p, -3/2]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Simp[c*(p/(e^2*(m + p + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & & IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(828\) vs. \(2(245)=490\).
Time = 4.04 (sec) , antiderivative size = 829, normalized size of antiderivative = 2.99
method | result | size |
default | \(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {2 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {8 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {10 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {14 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 d e c}-\frac {5 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 d e c}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{24 d e c}\right )}{2}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}}{e^{6}}\) | \(829\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^6,x,method=_RETURNVERB OSE)
Output:
1/e^6*(-2/3/(a*e^2-c*d^2)/(x+d/e)^6*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e) )^(9/2)+2*d*e*c/(a*e^2-c*d^2)*(-2/(a*e^2-c*d^2)/(x+d/e)^5*(d*e*c*(x+d/e)^2 +(a*e^2-c*d^2)*(x+d/e))^(9/2)+8*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2)/(x+d/ e)^4*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(9/2)-10*d*e*c/(a*e^2-c*d^2)* (2/3/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(9/2) -4*d*e*c/(a*e^2-c*d^2)*(2/5/(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^ 2-c*d^2)*(x+d/e))^(9/2)-14/5*d*e*c/(a*e^2-c*d^2)*(1/7*(d*e*c*(x+d/e)^2+(a* e^2-c*d^2)*(x+d/e))^(7/2)+1/2*(a*e^2-c*d^2)*(1/12*(2*d*e*c*(x+d/e)+a*e^2-c *d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-5/24*(a*e^2-c*d^ 2)^2/d/e/c*(1/8*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^ 2-c*d^2)*(x+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/d/e/c*(1/4*(2*d*e*c*(x+d/e)+a *e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2 -c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e* c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2))))))))))
Time = 0.77 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,x, algorithm=" fricas")
Output:
[1/48*(105*(c^3*d^7 - 2*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (c^3*d^5*e^2 - 2*a *c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 2*(c^3*d^6*e - 2*a*c^2*d^4*e^3 + a^2*c*d ^2*e^5)*x)*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*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(6*c^3*d^3*e^3 *x^3 - 105*c^3*d^6 + 175*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 8*a^3*e^6 - 3* (7*c^3*d^4*e^2 - 13*a*c^2*d^2*e^4)*x^2 - 2*(70*c^3*d^5*e - 119*a*c^2*d^3*e ^3 + 40*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^6* x^2 + 2*d*e^5*x + d^2*e^4), -1/24*(105*(c^3*d^7 - 2*a*c^2*d^5*e^2 + a^2*c* d^3*e^4 + (c^3*d^5*e^2 - 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 2*(c^3*d^6*e - 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*x)*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*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) - 2*(6*c^3*d^3*e^3* x^3 - 105*c^3*d^6 + 175*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 8*a^3*e^6 - 3*( 7*c^3*d^4*e^2 - 13*a*c^2*d^2*e^4)*x^2 - 2*(70*c^3*d^5*e - 119*a*c^2*d^3*e^ 3 + 40*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^6*x ^2 + 2*d*e^5*x + d^2*e^4)]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**6,x)
Output:
Integral(((d + e*x)*(a*e + c*d*x))**(7/2)/(d + e*x)**6, x)
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,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*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,x, algorithm=" giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{7/2}}{{\left (d+e\,x\right )}^6} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^6,x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^6, x)
Time = 0.31 (sec) , antiderivative size = 991, normalized size of antiderivative = 3.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,x)
Output:
( - 64*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*e**7 - 448*sqrt(d + e*x)*sqrt( a*e + c*d*x)*a**2*c*d**2*e**5 - 640*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c *d*e**6*x + 1400*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**3 + 1904*s qrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x + 312*sqrt(d + e*x)*sqrt (a*e + c*d*x)*a*c**2*d**2*e**5*x**2 - 840*sqrt(d + e*x)*sqrt(a*e + c*d*x)* c**3*d**6*e - 1120*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e**2*x - 168* sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**3*x**2 + 48*sqrt(d + e*x)*sqr t(a*e + c*d*x)*c**3*d**3*e**4*x**3 + 840*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**2*c*d**3*e**4 + 1680*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**2*c*d* *2*e**5*x + 840*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + s qrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d*e**6*x**2 - 1680*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*c**2*d**5*e**2 - 3360*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*c**2*d**4*e**3*x - 1680*sqrt(e)*sqrt(d)*sqr t(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*c**2*d**3*e**4*x**2 + 840*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...