\(\int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [98]

Optimal result

Integrand size = 48, antiderivative size = 341 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^{7/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 g \sqrt {d+e x} (f+g x)^{5/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4 \sqrt {d+e x}}+\frac {35 g^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3 \sqrt {d+e x}}+\frac {35 g^{3/2} (c d f-a e g)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{9/2} d^{9/2}} \] Output:

-2/3*(e*x+d)^(3/2)*(g*x+f)^(7/2)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/ 
2)-14/3*g*(e*x+d)^(1/2)*(g*x+f)^(5/2)/c^2/d^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/2)+35/4*g^2*(-a*e*g+c*d*f)*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c 
*d*e*x^2)^(1/2)/c^4/d^4/(e*x+d)^(1/2)+35/6*g^2*(g*x+f)^(3/2)*(a*d*e+(a*e^2 
+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/(e*x+d)^(1/2)+35/4*g^(3/2)*(-a*e*g+c*d* 
f)^2*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)*(g*x+f)^(1/2)/g^(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)
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.70 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{5/2} \left (-\sqrt {c} \sqrt {d} (a e+c d x) \sqrt {f+g x} \left (105 a^3 e^3 g^3+35 a^2 c d e^2 g^2 (-5 f+4 g x)+7 a c^2 d^2 e g \left (8 f^2-34 f g x+3 g^2 x^2\right )+c^3 d^3 \left (8 f^3+80 f^2 g x-39 f g^2 x^2-6 g^3 x^3\right )\right )+105 g^{3/2} (c d f-a e g)^2 (a e+c d x)^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{12 c^{9/2} d^{9/2} ((a e+c d x) (d+e x))^{5/2}} \] Input:

Integrate[((d + e*x)^(5/2)*(f + g*x)^(7/2))/(a*d*e + (c*d^2 + a*e^2)*x + c 
*d*e*x^2)^(5/2),x]
 

Output:

((d + e*x)^(5/2)*(-(Sqrt[c]*Sqrt[d]*(a*e + c*d*x)*Sqrt[f + g*x]*(105*a^3*e 
^3*g^3 + 35*a^2*c*d*e^2*g^2*(-5*f + 4*g*x) + 7*a*c^2*d^2*e*g*(8*f^2 - 34*f 
*g*x + 3*g^2*x^2) + c^3*d^3*(8*f^3 + 80*f^2*g*x - 39*f*g^2*x^2 - 6*g^3*x^3 
))) + 105*g^(3/2)*(c*d*f - a*e*g)^2*(a*e + c*d*x)^(5/2)*ArcTanh[(Sqrt[c]*S 
qrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(12*c^(9/2)*d^(9/2)*( 
(a*e + c*d*x)*(d + e*x))^(5/2))
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1251, 1251, 1253, 1253, 1268, 66, 221}

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.

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

\(\Big \downarrow \) 1251

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

\(\Big \downarrow \) 1251

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

\(\Big \downarrow \) 1253

\(\displaystyle \frac {7 g \left (\frac {5 g \left (\frac {3 (c d f-a e g) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d}-\frac {2 (d+e x)^{3/2} (f+g x)^{7/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {7 g \left (\frac {5 g \left (\frac {3 (c d f-a e g) \left (\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{c d}-\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d}-\frac {2 (d+e x)^{3/2} (f+g x)^{7/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

Int[((d + e*x)^(5/2)*(f + g*x)^(7/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x 
^2)^(5/2),x]
 

Output:

(-2*(d + e*x)^(3/2)*(f + g*x)^(7/2))/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c 
*d*e*x^2)^(3/2)) + (7*g*((-2*Sqrt[d + e*x]*(f + g*x)^(5/2))/(c*d*Sqrt[a*d* 
e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*g*(((f + g*x)^(3/2)*Sqrt[a*d*e + 
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*c*d*Sqrt[d + e*x]) + (3*(c*d*f - a*e*g) 
*((Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*Sqrt[d 
+ e*x]) + ((c*d*f - a*e*g)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g 
]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(c^(3/2)*d^(3/2)*Sq 
rt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/(4*c*d)))/(c*d)))/(3* 
c*d)
 

Defintions of rubi rules used

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1251

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*(f + g*x)^n*((a 
 + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Simp[e*g*(n/(c*(p + 1)))   Int[( 
d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 
 0] && LtQ[p, -1] && GtQ[n, 0]
 

rule 1253

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n* 
((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Simp[n*((c*e*f + c*d*g - 
b*e*g)/(c*e*(m - n - 1)))   Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c* 
x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d* 
e + a*e^2, 0] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (Intege 
rQ[2*p] || IntegerQ[n])
 

rule 1268

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d 
 + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f 
 + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1009\) vs. \(2(289)=578\).

Time = 2.66 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1010\)

Input:

int((e*x+d)^(5/2)*(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x, 
method=_RETURNVERBOSE)
 

Output:

1/24*(105*ln(1/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d 
*g)^(1/2))/(c*d*g)^(1/2))*a^2*c^2*d^2*e^2*g^4*x^2-210*ln(1/2*(2*c*d*g*x+a* 
e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a*c^ 
3*d^3*e*f*g^3*x^2+105*ln(1/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f) 
)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*c^4*d^4*f^2*g^2*x^2+210*ln(1/2*(2*c* 
d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/ 
2))*a^3*c*d*e^3*g^4*x-420*ln(1/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g* 
x+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^2*c^2*d^2*e^2*f*g^3*x+210*ln(1 
/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2))/(c* 
d*g)^(1/2))*a*c^3*d^3*e*f^2*g^2*x+12*c^3*d^3*g^3*x^3*((c*d*x+a*e)*(g*x+f)) 
^(1/2)*(c*d*g)^(1/2)+105*ln(1/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x 
+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^4*e^4*g^4-210*ln(1/2*(2*c*d*g*x 
+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a 
^3*c*d*e^3*f*g^3+105*ln(1/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f)) 
^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^2*c^2*d^2*e^2*f^2*g^2-42*a*c^2*d^2* 
e*g^3*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)+78*c^3*d^3*f*g^2*x^2*( 
(c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)-280*(c*d*g)^(1/2)*((c*d*x+a*e)*(g 
*x+f))^(1/2)*a^2*c*d*e^2*g^3*x+476*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/ 
2)*a*c^2*d^2*e*f*g^2*x-160*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/2)*c^3*d 
^3*f^2*g*x-210*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/2)*a^3*e^3*g^3+35...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (289) = 578\).

Time = 0.97 (sec) , antiderivative size = 1413, normalized size of antiderivative = 4.14 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)*(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 
/2),x, algorithm="fricas")
 

Output:

[1/48*(4*(6*c^3*d^3*g^3*x^3 - 8*c^3*d^3*f^3 - 56*a*c^2*d^2*e*f^2*g + 175*a 
^2*c*d*e^2*f*g^2 - 105*a^3*e^3*g^3 + 3*(13*c^3*d^3*f*g^2 - 7*a*c^2*d^2*e*g 
^3)*x^2 - 2*(40*c^3*d^3*f^2*g - 119*a*c^2*d^2*e*f*g^2 + 70*a^2*c*d*e^2*g^3 
)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + 
f) + 105*(a^2*c^2*d^3*e^2*f^2*g - 2*a^3*c*d^2*e^3*f*g^2 + a^4*d*e^4*g^3 + 
(c^4*d^4*e*f^2*g - 2*a*c^3*d^3*e^2*f*g^2 + a^2*c^2*d^2*e^3*g^3)*x^3 + ((c^ 
4*d^5 + 2*a*c^3*d^3*e^2)*f^2*g - 2*(a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3)*f*g^2 
 + (a^2*c^2*d^3*e^2 + 2*a^3*c*d*e^4)*g^3)*x^2 + ((2*a*c^3*d^4*e + a^2*c^2* 
d^2*e^3)*f^2*g - 2*(2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*f*g^2 + (2*a^3*c*d^2* 
e^3 + a^4*e^5)*g^3)*x)*sqrt(g/(c*d))*log(-(8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f 
^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d 
*e^2)*g^2)*x^2 + 4*(2*c^2*d^2*g*x + c^2*d^2*f + a*c*d*e*g)*sqrt(c*d*e*x^2 
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(g/(c*d)) + ( 
c^2*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)* 
g^2)*x)/(e*x + d)))/(c^6*d^6*e*x^3 + a^2*c^4*d^5*e^2 + (c^6*d^7 + 2*a*c^5* 
d^5*e^2)*x^2 + (2*a*c^5*d^6*e + a^2*c^4*d^4*e^3)*x), 1/24*(2*(6*c^3*d^3*g^ 
3*x^3 - 8*c^3*d^3*f^3 - 56*a*c^2*d^2*e*f^2*g + 175*a^2*c*d*e^2*f*g^2 - 105 
*a^3*e^3*g^3 + 3*(13*c^3*d^3*f*g^2 - 7*a*c^2*d^2*e*g^3)*x^2 - 2*(40*c^3*d^ 
3*f^2*g - 119*a*c^2*d^2*e*f*g^2 + 70*a^2*c*d*e^2*g^3)*x)*sqrt(c*d*e*x^2 + 
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) - 105*(a^2*c^2*d...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(5/2)*(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x** 
2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {7}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(5/2)*(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 
/2),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(5/2)*(g*x + f)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a* 
e^2)*x)^(5/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (289) = 578\).

Time = 0.77 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.00 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left ({\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left (3 \, {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left (\frac {2 \, {\left (c^{7} d^{7} e^{2} f g^{5} - a c^{6} d^{6} e^{3} g^{6}\right )} {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )}}{c^{8} d^{8} e^{6} f g {\left | g \right |} - a c^{7} d^{7} e^{7} g^{2} {\left | g \right |}} + \frac {7 \, {\left (c^{7} d^{7} e^{4} f^{2} g^{5} - 2 \, a c^{6} d^{6} e^{5} f g^{6} + a^{2} c^{5} d^{5} e^{6} g^{7}\right )}}{c^{8} d^{8} e^{6} f g {\left | g \right |} - a c^{7} d^{7} e^{7} g^{2} {\left | g \right |}}\right )} - \frac {140 \, {\left (c^{7} d^{7} e^{6} f^{3} g^{5} - 3 \, a c^{6} d^{6} e^{7} f^{2} g^{6} + 3 \, a^{2} c^{5} d^{5} e^{8} f g^{7} - a^{3} c^{4} d^{4} e^{9} g^{8}\right )}}{c^{8} d^{8} e^{6} f g {\left | g \right |} - a c^{7} d^{7} e^{7} g^{2} {\left | g \right |}}\right )} + \frac {105 \, {\left (c^{7} d^{7} e^{8} f^{4} g^{5} - 4 \, a c^{6} d^{6} e^{9} f^{3} g^{6} + 6 \, a^{2} c^{5} d^{5} e^{10} f^{2} g^{7} - 4 \, a^{3} c^{4} d^{4} e^{11} f g^{8} + a^{4} c^{3} d^{3} e^{12} g^{9}\right )}}{c^{8} d^{8} e^{6} f g {\left | g \right |} - a c^{7} d^{7} e^{7} g^{2} {\left | g \right |}}\right )}}{12 \, {\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )}^{2}} - \frac {35 \, {\left (c^{2} d^{2} f^{2} g^{3} - 2 \, a c d e f g^{4} + a^{2} e^{2} g^{5}\right )} \log \left ({\left | -\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} + \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \right |}\right )}{4 \, \sqrt {c d g} c^{4} d^{4} {\left | g \right |}} \] Input:

integrate((e*x+d)^(5/2)*(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5 
/2),x, algorithm="giac")
 

Output:

1/12*sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g 
)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*((e^2*f + (e*x + d)*e*g - d*e*g)*(3* 
(e^2*f + (e*x + d)*e*g - d*e*g)*(2*(c^7*d^7*e^2*f*g^5 - a*c^6*d^6*e^3*g^6) 
*(e^2*f + (e*x + d)*e*g - d*e*g)/(c^8*d^8*e^6*f*g*abs(g) - a*c^7*d^7*e^7*g 
^2*abs(g)) + 7*(c^7*d^7*e^4*f^2*g^5 - 2*a*c^6*d^6*e^5*f*g^6 + a^2*c^5*d^5* 
e^6*g^7)/(c^8*d^8*e^6*f*g*abs(g) - a*c^7*d^7*e^7*g^2*abs(g))) - 140*(c^7*d 
^7*e^6*f^3*g^5 - 3*a*c^6*d^6*e^7*f^2*g^6 + 3*a^2*c^5*d^5*e^8*f*g^7 - a^3*c 
^4*d^4*e^9*g^8)/(c^8*d^8*e^6*f*g*abs(g) - a*c^7*d^7*e^7*g^2*abs(g))) + 105 
*(c^7*d^7*e^8*f^4*g^5 - 4*a*c^6*d^6*e^9*f^3*g^6 + 6*a^2*c^5*d^5*e^10*f^2*g 
^7 - 4*a^3*c^4*d^4*e^11*f*g^8 + a^4*c^3*d^3*e^12*g^9)/(c^8*d^8*e^6*f*g*abs 
(g) - a*c^7*d^7*e^7*g^2*abs(g)))/(c*d*e^2*f*g - a*e^3*g^2 - (e^2*f + (e*x 
+ d)*e*g - d*e*g)*c*d*g)^2 - 35/4*(c^2*d^2*f^2*g^3 - 2*a*c*d*e*f*g^4 + a^2 
*e^2*g^5)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt( 
-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt( 
c*d*g)*c^4*d^4*abs(g))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{7/2}\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int(((f + g*x)^(7/2)*(d + e*x)^(5/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x 
^2)^(5/2),x)
 

Output:

int(((f + g*x)^(7/2)*(d + e*x)^(5/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x 
^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(5/2)*(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

(840*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(g)*sqrt(a*e + c*d 
*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a**3*e**3*g**3 - 
 1680*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(g)*sqrt(a*e + c* 
d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a**2*c*d*e**2*f 
*g**2 + 840*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(g)*sqrt(a* 
e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a**2*c*d* 
e**2*g**3*x + 840*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(g)*s 
qrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a*c 
**2*d**2*e*f**2*g - 1680*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sq 
rt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d* 
f))*a*c**2*d**2*e*f*g**2*x + 840*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x) 
*log((sqrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e* 
g - c*d*f))*c**3*d**3*f**2*g*x + 175*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c* 
d*x)*a**3*e**3*g**3 - 350*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c 
*d*e**2*f*g**2 + 175*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d*e* 
*2*g**3*x + 175*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**2*e*f* 
*2*g - 350*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**2*e*f*g**2* 
x + 175*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**3*f**2*g*x - 840 
*sqrt(f + g*x)*a**3*c*d*e**3*g**3 + 1400*sqrt(f + g*x)*a**2*c**2*d**2*e**2 
*f*g**2 - 1120*sqrt(f + g*x)*a**2*c**2*d**2*e**2*g**3*x - 448*sqrt(f + ...