\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{d+e x} \, dx\) [236]

Optimal result

Integrand size = 37, antiderivative size = 496 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=-\frac {5 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^3 d^3 e^4}+\frac {5 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1536 c^3 d^3 e^3 (d+e x)}-\frac {\left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{384 c^3 d^3 e^2 (d+e x)^2}+\frac {\left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{448 c^3 d^3 e (d+e x)^3}+\frac {\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{56 c^3 d^3 (d+e x)^4}+\frac {5 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{84 c^2 d^2 (d+e x)^3}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{7 c d (d+e x)^2}+\frac {5 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}} \] Output:

-5/1024*(-a*e^2+c*d^2)^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e 
^4+5/1536*(-a*e^2+c*d^2)^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3 
/e^3/(e*x+d)-1/384*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2 
)/c^3/d^3/e^2/(e*x+d)^2+1/448*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d* 
e*x^2)^(7/2)/c^3/d^3/e/(e*x+d)^3+1/56*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2 
)*x+c*d*e*x^2)^(9/2)/c^3/d^3/(e*x+d)^4+5/84*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c 
*d^2)*x+c*d*e*x^2)^(9/2)/c^2/d^2/(e*x+d)^3+1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d* 
e*x^2)^(9/2)/c/d/(e*x+d)^2+5/1024*(-a*e^2+c*d^2)^7*arctanh(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))/c^(7/2)/d^(7/2)/ 
e^(9/2)
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (105 a^6 e^{12}-70 a^5 c d e^{10} (10 d+e x)+7 a^4 c^2 d^2 e^8 \left (283 d^2+66 d e x+8 e^2 x^2\right )+4 a^3 c^3 d^3 e^6 \left (768 d^3+4285 d^2 e x+4516 d e^2 x^2+1524 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (-1981 d^4+1292 d^3 e x+27648 d^2 e^2 x^2+36496 d e^3 x^3+13696 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (350 d^5-231 d^4 e x+184 d^3 e^2 x^2+9400 d^2 e^3 x^3+13824 d e^4 x^4+5504 e^5 x^5\right )+c^6 d^6 \left (-105 d^6+70 d^5 e x-56 d^4 e^2 x^2+48 d^3 e^3 x^3+4736 d^2 e^4 x^4+7424 d e^5 x^5+3072 e^6 x^6\right )\right )}{(a e+c d x)^2 (d+e x)^2}+\frac {105 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{21504 c^{7/2} d^{7/2} e^{9/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x),x]
 

Output:

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(105*a^6*e^12 - 
 70*a^5*c*d*e^10*(10*d + e*x) + 7*a^4*c^2*d^2*e^8*(283*d^2 + 66*d*e*x + 8* 
e^2*x^2) + 4*a^3*c^3*d^3*e^6*(768*d^3 + 4285*d^2*e*x + 4516*d*e^2*x^2 + 15 
24*e^3*x^3) + a^2*c^4*d^4*e^4*(-1981*d^4 + 1292*d^3*e*x + 27648*d^2*e^2*x^ 
2 + 36496*d*e^3*x^3 + 13696*e^4*x^4) + 2*a*c^5*d^5*e^2*(350*d^5 - 231*d^4* 
e*x + 184*d^3*e^2*x^2 + 9400*d^2*e^3*x^3 + 13824*d*e^4*x^4 + 5504*e^5*x^5) 
 + c^6*d^6*(-105*d^6 + 70*d^5*e*x - 56*d^4*e^2*x^2 + 48*d^3*e^3*x^3 + 4736 
*d^2*e^4*x^4 + 7424*d*e^5*x^5 + 3072*e^6*x^6)))/((a*e + c*d*x)^2*(d + e*x) 
^2) + (105*(c*d^2 - a*e^2)^7*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt 
[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(21504*c^ 
(7/2)*d^(7/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1131, 1087, 1087, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{d+e x} \, dx\)

\(\Big \downarrow \) 1131

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 e}-\frac {\left (c d^2-a e^2\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{2 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{24 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{24 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 e}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x),x]
 

Output:

(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(7*e) - ((c*d^2 - a*e^2)*((( 
c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/ 
(12*c*d*e) - (5*(c*d^2 - a*e^2)^2*(((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + ( 
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*c*d*e) - (3*(c*d^2 - a*e^2)^2*(((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*d*e)))/(24*c*d*e)))/(2*e)
 

Defintions of rubi rules used

rule 219

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])
 

rule 1087

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])
 

rule 1092

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]
 

rule 1131

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 + 2*p + 1))), x 
] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1)))   Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne 
Q[m + 2*p + 1, 0] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 d e c}-\frac {5 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 d e c}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{24 d e c}\right )}{2}}{e}\)	\(424\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d),x,method=_RETURNVERBOS 
E)
 

Output:

1/e*(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)))))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1270, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d),x, algorithm="fr 
icas")
 

Output:

[-1/86016*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3 
*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 
 - a^7*e^14)*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*(3072*c^7*d^7*e^ 
7*x^6 - 105*c^7*d^13*e + 700*a*c^6*d^11*e^3 - 1981*a^2*c^5*d^9*e^5 + 3072* 
a^3*c^4*d^7*e^7 + 1981*a^4*c^3*d^5*e^9 - 700*a^5*c^2*d^3*e^11 + 105*a^6*c* 
d*e^13 + 256*(29*c^7*d^8*e^6 + 43*a*c^6*d^6*e^8)*x^5 + 128*(37*c^7*d^9*e^5 
 + 216*a*c^6*d^7*e^7 + 107*a^2*c^5*d^5*e^9)*x^4 + 16*(3*c^7*d^10*e^4 + 117 
5*a*c^6*d^8*e^6 + 2281*a^2*c^5*d^6*e^8 + 381*a^3*c^4*d^4*e^10)*x^3 - 8*(7* 
c^7*d^11*e^3 - 46*a*c^6*d^9*e^5 - 3456*a^2*c^5*d^7*e^7 - 2258*a^3*c^4*d^5* 
e^9 - 7*a^4*c^3*d^3*e^11)*x^2 + 2*(35*c^7*d^12*e^2 - 231*a*c^6*d^10*e^4 + 
646*a^2*c^5*d^8*e^6 + 8570*a^3*c^4*d^6*e^8 + 231*a^4*c^3*d^4*e^10 - 35*a^5 
*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^ 
5), -1/43008*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35* 
a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e 
^12 - a^7*e^14)*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*(3072*c^7*d^7*e^7*x^6 - 105*c^7* 
d^13*e + 700*a*c^6*d^11*e^3 - 1981*a^2*c^5*d^9*e^5 + 3072*a^3*c^4*d^7*e...
 

Sympy [A] (verification not implemented)

Time = 17.79 (sec) , antiderivative size = 11608, normalized size of antiderivative = 23.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d),x)
 

Output:

a**3*d**2*e**3*Piecewise(((x/2 + (a*e**2/4 + c*d**2/4)/(c*d*e))*sqrt(a*d*e 
 + c*d*e*x**2 + x*(a*e**2 + c*d**2)) + (a*d*e/2 - (a*e**2/4 + c*d**2/4)*(a 
*e**2 + c*d**2)/(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), N 
e(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)), Ne(c*d*e, 0)), (2*(a*d*e + x*(a*e**2 + c*d 
**2))**(3/2)/(3*(a*e**2 + c*d**2)), Ne(a*e**2 + c*d**2, 0)), (x*sqrt(a*d*e 
), True)) + 2*a**3*d*e**4*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a* 
e**2 + c*d**2)*(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/ 
(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)) + (x**2/3 + x*(a*e**2/6 + c*d**2/6)/(2*c*d*e) + 
(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d 
*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)), Ne(c*d*e, 0)), (2*(-a 
*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 + (a*d*e + x*(a*e**2 + c*d**2) 
)**(5/2)/5)/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d* 
e)/2, True)) + a**3*e**5*Piecewise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d),x, algorithm="ma 
xima")
 

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
 

Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=\frac {1}{21504} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{3} d^{3} e^{2} x + \frac {29 \, c^{9} d^{10} e^{7} + 43 \, a c^{8} d^{8} e^{9}}{c^{6} d^{6} e^{6}}\right )} x + \frac {37 \, c^{9} d^{11} e^{6} + 216 \, a c^{8} d^{9} e^{8} + 107 \, a^{2} c^{7} d^{7} e^{10}}{c^{6} d^{6} e^{6}}\right )} x + \frac {3 \, c^{9} d^{12} e^{5} + 1175 \, a c^{8} d^{10} e^{7} + 2281 \, a^{2} c^{7} d^{8} e^{9} + 381 \, a^{3} c^{6} d^{6} e^{11}}{c^{6} d^{6} e^{6}}\right )} x - \frac {7 \, c^{9} d^{13} e^{4} - 46 \, a c^{8} d^{11} e^{6} - 3456 \, a^{2} c^{7} d^{9} e^{8} - 2258 \, a^{3} c^{6} d^{7} e^{10} - 7 \, a^{4} c^{5} d^{5} e^{12}}{c^{6} d^{6} e^{6}}\right )} x + \frac {35 \, c^{9} d^{14} e^{3} - 231 \, a c^{8} d^{12} e^{5} + 646 \, a^{2} c^{7} d^{10} e^{7} + 8570 \, a^{3} c^{6} d^{8} e^{9} + 231 \, a^{4} c^{5} d^{6} e^{11} - 35 \, a^{5} c^{4} d^{4} e^{13}}{c^{6} d^{6} e^{6}}\right )} x - \frac {105 \, c^{9} d^{15} e^{2} - 700 \, a c^{8} d^{13} e^{4} + 1981 \, a^{2} c^{7} d^{11} e^{6} - 3072 \, a^{3} c^{6} d^{9} e^{8} - 1981 \, a^{4} c^{5} d^{7} e^{10} + 700 \, a^{5} c^{4} d^{5} e^{12} - 105 \, a^{6} c^{3} d^{3} e^{14}}{c^{6} d^{6} e^{6}}\right )} - \frac {5 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2048 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d),x, algorithm="gi 
ac")
 

Output:

1/21504*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(12*c^3 
*d^3*e^2*x + (29*c^9*d^10*e^7 + 43*a*c^8*d^8*e^9)/(c^6*d^6*e^6))*x + (37*c 
^9*d^11*e^6 + 216*a*c^8*d^9*e^8 + 107*a^2*c^7*d^7*e^10)/(c^6*d^6*e^6))*x + 
 (3*c^9*d^12*e^5 + 1175*a*c^8*d^10*e^7 + 2281*a^2*c^7*d^8*e^9 + 381*a^3*c^ 
6*d^6*e^11)/(c^6*d^6*e^6))*x - (7*c^9*d^13*e^4 - 46*a*c^8*d^11*e^6 - 3456* 
a^2*c^7*d^9*e^8 - 2258*a^3*c^6*d^7*e^10 - 7*a^4*c^5*d^5*e^12)/(c^6*d^6*e^6 
))*x + (35*c^9*d^14*e^3 - 231*a*c^8*d^12*e^5 + 646*a^2*c^7*d^10*e^7 + 8570 
*a^3*c^6*d^8*e^9 + 231*a^4*c^5*d^6*e^11 - 35*a^5*c^4*d^4*e^13)/(c^6*d^6*e^ 
6))*x - (105*c^9*d^15*e^2 - 700*a*c^8*d^13*e^4 + 1981*a^2*c^7*d^11*e^6 - 3 
072*a^3*c^6*d^9*e^8 - 1981*a^4*c^5*d^7*e^10 + 700*a^5*c^4*d^5*e^12 - 105*a 
^6*c^3*d^3*e^14)/(c^6*d^6*e^6)) - 5/2048*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21 
*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d 
^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d 
*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt( 
c*d*e)*c^3*d^3*e^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{7/2}}{d+e\,x} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 1308, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d),x)
 

Output:

(105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*c*d*e**13 - 700*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**5*c**2*d**3*e**11 - 70*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**5*c**2*d**2*e**12*x + 1981*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3*d* 
*5*e**9 + 462*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3*d**4*e**10*x + 56* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3*d**3*e**11*x**2 + 3072*sqrt(d + 
e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d**7*e**7 + 17140*sqrt(d + e*x)*sqrt(a*e 
+ c*d*x)*a**3*c**4*d**6*e**8*x + 18064*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a** 
3*c**4*d**5*e**9*x**2 + 6096*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d** 
4*e**10*x**3 - 1981*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**9*e**5 + 
1292*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**8*e**6*x + 27648*sqrt(d 
+ e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**7*e**7*x**2 + 36496*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**2*c**5*d**6*e**8*x**3 + 13696*sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*a**2*c**5*d**5*e**9*x**4 + 700*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c* 
*6*d**11*e**3 - 462*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6*d**10*e**4*x + 
368*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6*d**9*e**5*x**2 + 18800*sqrt(d + 
 e*x)*sqrt(a*e + c*d*x)*a*c**6*d**8*e**6*x**3 + 27648*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a*c**6*d**7*e**7*x**4 + 11008*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a*c**6*d**6*e**8*x**5 - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**7*d**13*e + 
 70*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**7*d**12*e**2*x - 56*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*c**7*d**11*e**3*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d...