\(\int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx\) [1184]

Optimal result

Integrand size = 26, antiderivative size = 203 \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=-\frac {2 f (2 b d e+b c f-3 a d f) \sqrt {c+d x}}{b^4 \sqrt {a+b x}}+\frac {d f^2 \sqrt {a+b x} \sqrt {c+d x}}{b^4}-\frac {4 f (b e-a f) (c+d x)^{3/2}}{3 b^3 (a+b x)^{3/2}}-\frac {2 (b e-a f)^2 (c+d x)^{5/2}}{5 b^2 (b c-a d) (a+b x)^{5/2}}+\frac {\sqrt {d} f (4 b d e+3 b c f-7 a d f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{9/2}} \] Output:

-2*f*(-3*a*d*f+b*c*f+2*b*d*e)*(d*x+c)^(1/2)/b^4/(b*x+a)^(1/2)+d*f^2*(b*x+a 
)^(1/2)*(d*x+c)^(1/2)/b^4-4/3*f*(-a*f+b*e)*(d*x+c)^(3/2)/b^3/(b*x+a)^(3/2) 
-2/5*(-a*f+b*e)^2*(d*x+c)^(5/2)/b^2/(-a*d+b*c)/(b*x+a)^(5/2)+d^(1/2)*f*(-7 
*a*d*f+3*b*c*f+4*b*d*e)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2 
))/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.35 \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=-\frac {\sqrt {c+d x} \left (105 a^4 d^2 f^2+5 a^3 b d f (-12 d e-23 c f+49 d f x)+a b^3 f \left (c d x (96 e-185 f x)+8 c^2 (e+5 f x)+d^2 x^2 (-92 e+15 f x)\right )+a^2 b^2 f \left (16 c^2 f+c d (40 e-273 f x)+7 d^2 x (-20 e+23 f x)\right )+b^4 \left (6 d^2 e^2 x^2+c d x \left (12 e^2+80 e f x-15 f^2 x^2\right )+c^2 \left (6 e^2+20 e f x+30 f^2 x^2\right )\right )\right )}{15 b^4 (b c-a d) (a+b x)^{5/2}}+\frac {\sqrt {d} f (4 b d e+3 b c f-7 a d f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{9/2}} \] Input:

Integrate[((c + d*x)^(3/2)*(e + f*x)^2)/(a + b*x)^(7/2),x]
 

Output:

-1/15*(Sqrt[c + d*x]*(105*a^4*d^2*f^2 + 5*a^3*b*d*f*(-12*d*e - 23*c*f + 49 
*d*f*x) + a*b^3*f*(c*d*x*(96*e - 185*f*x) + 8*c^2*(e + 5*f*x) + d^2*x^2*(- 
92*e + 15*f*x)) + a^2*b^2*f*(16*c^2*f + c*d*(40*e - 273*f*x) + 7*d^2*x*(-2 
0*e + 23*f*x)) + b^4*(6*d^2*e^2*x^2 + c*d*x*(12*e^2 + 80*e*f*x - 15*f^2*x^ 
2) + c^2*(6*e^2 + 20*e*f*x + 30*f^2*x^2))))/(b^4*(b*c - a*d)*(a + b*x)^(5/ 
2)) + (Sqrt[d]*f*(4*b*d*e + 3*b*c*f - 7*a*d*f)*ArcTanh[(Sqrt[d]*Sqrt[a + b 
*x])/(Sqrt[b]*Sqrt[c + d*x])])/b^(9/2)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 27, 87, 57, 60, 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.

\(\displaystyle \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx\)

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)^(3/2)*(e + f*x)^2)/(a + b*x)^(7/2),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

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 87

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

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1821\) vs. \(2(171)=342\).

Time = 0.31 (sec) , antiderivative size = 1822, normalized size of antiderivative = 8.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1822\)

Input:

int((d*x+c)^(3/2)*(f*x+e)^2/(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30*(d*x+c)^(1/2)*(-450*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^ 
(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*b^2*c*d^2*f^2*x-180*ln(1/2*(2*b*d*x+2*((b* 
x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*b^2*d^3*e*f*x-30 
*a*b^3*d^2*f^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+30*b^4*c*d*f^2*x^3* 
((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+184*a*b^3*d^2*e*f*x^2*((b*x+a)*(d*x+c) 
)^(1/2)*(d*b)^(1/2)-160*b^4*c*d*e*f*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2 
)+546*a^2*b^2*c*d*f^2*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+280*a^2*b^2*d^ 
2*e*f*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-80*a^2*b^2*c*d*e*f*((b*x+a)*(d 
*x+c))^(1/2)*(d*b)^(1/2)+180*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d* 
b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^4*c*d^2*e*f*x^2+180*ln(1/2*(2*b*d*x+2*( 
(b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*b^3*c*d^2*e*f 
*x-192*a*b^3*c*d*e*f*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-450*ln(1/2*(2*b 
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*b^3*c 
*d^2*f^2*x^2-180*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d 
+b*c)/(d*b)^(1/2))*a^2*b^3*d^3*e*f*x^2+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x 
+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^4*c^2*d*f^2*x^2+135*ln(1/ 
2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2 
*b^3*c^2*d*f^2*x+60*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+ 
a*d+b*c)/(d*b)^(1/2))*a^3*b^2*c*d^2*e*f-322*a^2*b^2*d^2*f^2*x^2*((b*x+a)*( 
d*x+c))^(1/2)*(d*b)^(1/2)-490*a^3*b*d^2*f^2*x*((b*x+a)*(d*x+c))^(1/2)*(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (171) = 342\).

Time = 3.18 (sec) , antiderivative size = 1347, normalized size of antiderivative = 6.64 \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)*(f*x+e)^2/(b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

[-1/60*(15*((4*(b^5*c*d - a*b^4*d^2)*e*f + (3*b^5*c^2 - 10*a*b^4*c*d + 7*a 
^2*b^3*d^2)*f^2)*x^3 + 4*(a^3*b^2*c*d - a^4*b*d^2)*e*f + (3*a^3*b^2*c^2 - 
10*a^4*b*c*d + 7*a^5*d^2)*f^2 + 3*(4*(a*b^4*c*d - a^2*b^3*d^2)*e*f + (3*a* 
b^4*c^2 - 10*a^2*b^3*c*d + 7*a^3*b^2*d^2)*f^2)*x^2 + 3*(4*(a^2*b^3*c*d - a 
^3*b^2*d^2)*e*f + (3*a^2*b^3*c^2 - 10*a^3*b^2*c*d + 7*a^4*b*d^2)*f^2)*x)*s 
qrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b^2*d*x 
+ b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b* 
d^2)*x) + 4*(6*b^4*c^2*e^2 - 15*(b^4*c*d - a*b^3*d^2)*f^2*x^3 + 4*(2*a*b^3 
*c^2 + 10*a^2*b^2*c*d - 15*a^3*b*d^2)*e*f + (16*a^2*b^2*c^2 - 115*a^3*b*c* 
d + 105*a^4*d^2)*f^2 + (6*b^4*d^2*e^2 + 4*(20*b^4*c*d - 23*a*b^3*d^2)*e*f 
+ (30*b^4*c^2 - 185*a*b^3*c*d + 161*a^2*b^2*d^2)*f^2)*x^2 + (12*b^4*c*d*e^ 
2 + 4*(5*b^4*c^2 + 24*a*b^3*c*d - 35*a^2*b^2*d^2)*e*f + (40*a*b^3*c^2 - 27 
3*a^2*b^2*c*d + 245*a^3*b*d^2)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*b 
^5*c - a^4*b^4*d + (b^8*c - a*b^7*d)*x^3 + 3*(a*b^7*c - a^2*b^6*d)*x^2 + 3 
*(a^2*b^6*c - a^3*b^5*d)*x), -1/30*(15*((4*(b^5*c*d - a*b^4*d^2)*e*f + (3* 
b^5*c^2 - 10*a*b^4*c*d + 7*a^2*b^3*d^2)*f^2)*x^3 + 4*(a^3*b^2*c*d - a^4*b* 
d^2)*e*f + (3*a^3*b^2*c^2 - 10*a^4*b*c*d + 7*a^5*d^2)*f^2 + 3*(4*(a*b^4*c* 
d - a^2*b^3*d^2)*e*f + (3*a*b^4*c^2 - 10*a^2*b^3*c*d + 7*a^3*b^2*d^2)*f^2) 
*x^2 + 3*(4*(a^2*b^3*c*d - a^3*b^2*d^2)*e*f + (3*a^2*b^3*c^2 - 10*a^3*b^2* 
c*d + 7*a^4*b*d^2)*f^2)*x)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*...
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}} \left (e + f x\right )^{2}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((d*x+c)**(3/2)*(f*x+e)**2/(b*x+a)**(7/2),x)
 

Output:

Integral((c + d*x)**(3/2)*(e + f*x)**2/(a + b*x)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)*(f*x+e)^2/(b*x+a)^(7/2),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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2696 vs. \(2 (171) = 342\).

Time = 0.69 (sec) , antiderivative size = 2696, normalized size of antiderivative = 13.28 \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)*(f*x+e)^2/(b*x+a)^(7/2),x, algorithm="giac")
 

Output:

sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*d*f^2*abs(b)/b^6 - 1/2*( 
4*sqrt(b*d)*b*d*e*f*abs(b) + 3*sqrt(b*d)*b*c*f^2*abs(b) - 7*sqrt(b*d)*a*d* 
f^2*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* 
b*d))^2)/b^6 - 4/15*(3*sqrt(b*d)*b^10*c^4*d^2*e^2*abs(b) - 12*sqrt(b*d)*a* 
b^9*c^3*d^3*e^2*abs(b) + 18*sqrt(b*d)*a^2*b^8*c^2*d^4*e^2*abs(b) - 12*sqrt 
(b*d)*a^3*b^7*c*d^5*e^2*abs(b) + 3*sqrt(b*d)*a^4*b^6*d^6*e^2*abs(b) + 40*s 
qrt(b*d)*b^10*c^5*d*e*f*abs(b) - 206*sqrt(b*d)*a*b^9*c^4*d^2*e*f*abs(b) + 
424*sqrt(b*d)*a^2*b^8*c^3*d^3*e*f*abs(b) - 436*sqrt(b*d)*a^3*b^7*c^2*d^4*e 
*f*abs(b) + 224*sqrt(b*d)*a^4*b^6*c*d^5*e*f*abs(b) - 46*sqrt(b*d)*a^5*b^5* 
d^6*e*f*abs(b) + 15*sqrt(b*d)*b^10*c^6*f^2*abs(b) - 130*sqrt(b*d)*a*b^9*c^ 
5*d*f^2*abs(b) + 428*sqrt(b*d)*a^2*b^8*c^4*d^2*f^2*abs(b) - 712*sqrt(b*d)* 
a^3*b^7*c^3*d^3*f^2*abs(b) + 643*sqrt(b*d)*a^4*b^6*c^2*d^4*f^2*abs(b) - 30 
2*sqrt(b*d)*a^5*b^5*c*d^5*f^2*abs(b) + 58*sqrt(b*d)*a^6*b^4*d^6*f^2*abs(b) 
 - 140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a 
*b*d))^2*b^8*c^4*d*e*f*abs(b) + 560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s 
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7*c^3*d^2*e*f*abs(b) - 840*sqrt( 
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2 
*b^6*c^2*d^3*e*f*abs(b) + 560*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^ 
2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*c*d^4*e*f*abs(b) - 140*sqrt(b*d)*( 
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx=\int \frac {{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^{7/2}} \,d x \] Input:

int(((e + f*x)^2*(c + d*x)^(3/2))/(a + b*x)^(7/2),x)
 

Output:

int(((e + f*x)^2*(c + d*x)^(3/2))/(a + b*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 1724, normalized size of antiderivative = 8.49 \[ \int \frac {(c+d x)^{3/2} (e+f x)^2}{(a+b x)^{7/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(3/2)*(f*x+e)^2/(b*x+a)^(7/2),x)
 

Output:

( - 420*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b) 
*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d**2*f**2 + 600*sqrt(d)*sqrt(b)*sqrt 
(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b 
*c))*a**3*b*c*d*f**2 + 240*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt 
(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**3*b*d**2*e*f - 840* 
sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
+ d*x))/sqrt(a*d - b*c))*a**3*b*d**2*f**2*x - 180*sqrt(d)*sqrt(b)*sqrt(a + 
 b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c)) 
*a**2*b**2*c**2*f**2 - 240*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt 
(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*c*d*e*f + 12 
00*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt 
(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*c*d*f**2*x + 480*sqrt(d)*sqrt(b)*sqr 
t(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - 
b*c))*a**2*b**2*d**2*e*f*x - 420*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d 
)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*d**2*f 
**2*x**2 - 360*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + 
sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**3*c**2*f**2*x - 480*sqrt(d)*s 
qrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/s 
qrt(a*d - b*c))*a*b**3*c*d*e*f*x + 600*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log(( 
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**3*...