\(\int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx\) [213]

Optimal result

Integrand size = 24, antiderivative size = 251 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx=-\frac {2 (B d-A e) \sqrt {a+b x}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{105 e (b d-a e)^3 (d+e x)^{5/2}}+\frac {16 b^2 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{315 e (b d-a e)^4 (d+e x)^{3/2}}+\frac {32 b^3 (b B d+8 A b e-9 a B e) \sqrt {a+b x}}{315 e (b d-a e)^5 \sqrt {d+e x}} \] Output:

-2/9*(-A*e+B*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)/(e*x+d)^(9/2)+2/63*(8*A*b*e-9*B 
*a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^2/(e*x+d)^(7/2)+4/105*b*(8*A*b*e-9* 
B*a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^3/(e*x+d)^(5/2)+16/315*b^2*(8*A*b* 
e-9*B*a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^4/(e*x+d)^(3/2)+32/315*b^3*(8* 
A*b*e-9*B*a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^5/(e*x+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx=\frac {2 \sqrt {a+b x} \left (-35 B d e^3 (a+b x)^4+35 A e^4 (a+b x)^4+135 b B d e^2 (a+b x)^3 (d+e x)-180 A b e^3 (a+b x)^3 (d+e x)+45 a B e^3 (a+b x)^3 (d+e x)-189 b^2 B d e (a+b x)^2 (d+e x)^2+378 A b^2 e^2 (a+b x)^2 (d+e x)^2-189 a b B e^2 (a+b x)^2 (d+e x)^2+105 b^3 B d (a+b x) (d+e x)^3-420 A b^3 e (a+b x) (d+e x)^3+315 a b^2 B e (a+b x) (d+e x)^3+315 A b^4 (d+e x)^4-315 a b^3 B (d+e x)^4\right )}{315 (b d-a e)^5 (d+e x)^{9/2}} \] Input:

Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]
 

Output:

(2*Sqrt[a + b*x]*(-35*B*d*e^3*(a + b*x)^4 + 35*A*e^4*(a + b*x)^4 + 135*b*B 
*d*e^2*(a + b*x)^3*(d + e*x) - 180*A*b*e^3*(a + b*x)^3*(d + e*x) + 45*a*B* 
e^3*(a + b*x)^3*(d + e*x) - 189*b^2*B*d*e*(a + b*x)^2*(d + e*x)^2 + 378*A* 
b^2*e^2*(a + b*x)^2*(d + e*x)^2 - 189*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 + 
105*b^3*B*d*(a + b*x)*(d + e*x)^3 - 420*A*b^3*e*(a + b*x)*(d + e*x)^3 + 31 
5*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 315*A*b^4*(d + e*x)^4 - 315*a*b^3*B*(d 
 + e*x)^4))/(315*(b*d - a*e)^5*(d + e*x)^(9/2))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {87, 55, 55, 55, 48}

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.

\(\displaystyle \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-9 a B e+8 A b e+b B d) \int \frac {1}{\sqrt {a+b x} (d+e x)^{9/2}}dx}{9 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-9 a B e+8 A b e+b B d) \left (\frac {6 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{7/2}}dx}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-9 a B e+8 A b e+b B d) \left (\frac {6 b \left (\frac {4 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}}dx}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {(-9 a B e+8 A b e+b B d) \left (\frac {6 b \left (\frac {4 b \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}}dx}{3 (b d-a e)}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right )}{9 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {6 b \left (\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {d+e x} (b d-a e)^2}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 (b d-a e)}+\frac {2 \sqrt {a+b x}}{5 (d+e x)^{5/2} (b d-a e)}\right )}{7 (b d-a e)}+\frac {2 \sqrt {a+b x}}{7 (d+e x)^{7/2} (b d-a e)}\right ) (-9 a B e+8 A b e+b B d)}{9 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\)

Input:

Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(11/2)),x]
 

Output:

(-2*(B*d - A*e)*Sqrt[a + b*x])/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + ((b*B*d 
 + 8*A*b*e - 9*a*B*e)*((2*Sqrt[a + b*x])/(7*(b*d - a*e)*(d + e*x)^(7/2)) + 
 (6*b*((2*Sqrt[a + b*x])/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (4*b*((2*Sqrt[a 
 + b*x])/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (4*b*Sqrt[a + b*x])/(3*(b*d - a 
*e)^2*Sqrt[d + e*x])))/(5*(b*d - a*e))))/(7*(b*d - a*e))))/(9*e*(b*d - a*e 
))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

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]))))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(221)=442\).

Time = 0.34 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.79

method	result	size
default	\(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} e^{4} x^{4}-144 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}+576 A \,b^{4} d \,e^{3} x^{3}+72 B \,a^{2} b^{2} e^{4} x^{3}-656 B a \,b^{3} d \,e^{3} x^{3}+72 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}-288 A a \,b^{3} d \,e^{3} x^{2}+1008 A \,b^{4} d^{2} e^{2} x^{2}-54 B \,a^{3} b \,e^{4} x^{2}+330 B \,a^{2} b^{2} d \,e^{3} x^{2}-1170 B a \,b^{3} d^{2} e^{2} x^{2}+126 B \,b^{4} d^{3} e \,x^{2}-40 A \,a^{3} b \,e^{4} x +216 A \,a^{2} b^{2} d \,e^{3} x -504 A a \,b^{3} d^{2} e^{2} x +840 A \,b^{4} d^{3} e x +45 B \,a^{4} e^{4} x -248 B \,a^{3} b d \,e^{3} x +594 B \,a^{2} b^{2} d^{2} e^{2} x -1008 B a \,b^{3} d^{3} e x +105 B \,b^{4} d^{4} x +35 A \,a^{4} e^{4}-180 A \,a^{3} b d \,e^{3}+378 A \,a^{2} b^{2} d^{2} e^{2}-420 A a \,b^{3} d^{3} e +315 A \,b^{4} d^{4}+10 B \,a^{4} d \,e^{3}-54 B \,a^{3} b \,d^{2} e^{2}+126 B \,a^{2} b^{2} d^{3} e -210 B a \,b^{3} d^{4}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a e -d b \right )^{5}}\)	\(449\)
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} e^{4} x^{4}-144 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}+576 A \,b^{4} d \,e^{3} x^{3}+72 B \,a^{2} b^{2} e^{4} x^{3}-656 B a \,b^{3} d \,e^{3} x^{3}+72 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}-288 A a \,b^{3} d \,e^{3} x^{2}+1008 A \,b^{4} d^{2} e^{2} x^{2}-54 B \,a^{3} b \,e^{4} x^{2}+330 B \,a^{2} b^{2} d \,e^{3} x^{2}-1170 B a \,b^{3} d^{2} e^{2} x^{2}+126 B \,b^{4} d^{3} e \,x^{2}-40 A \,a^{3} b \,e^{4} x +216 A \,a^{2} b^{2} d \,e^{3} x -504 A a \,b^{3} d^{2} e^{2} x +840 A \,b^{4} d^{3} e x +45 B \,a^{4} e^{4} x -248 B \,a^{3} b d \,e^{3} x +594 B \,a^{2} b^{2} d^{2} e^{2} x -1008 B a \,b^{3} d^{3} e x +105 B \,b^{4} d^{4} x +35 A \,a^{4} e^{4}-180 A \,a^{3} b d \,e^{3}+378 A \,a^{2} b^{2} d^{2} e^{2}-420 A a \,b^{3} d^{3} e +315 A \,b^{4} d^{4}+10 B \,a^{4} d \,e^{3}-54 B \,a^{3} b \,d^{2} e^{2}+126 B \,a^{2} b^{2} d^{3} e -210 B a \,b^{3} d^{4}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)
orering	\(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} e^{4} x^{4}-144 B a \,b^{3} e^{4} x^{4}+16 B \,b^{4} d \,e^{3} x^{4}-64 A a \,b^{3} e^{4} x^{3}+576 A \,b^{4} d \,e^{3} x^{3}+72 B \,a^{2} b^{2} e^{4} x^{3}-656 B a \,b^{3} d \,e^{3} x^{3}+72 B \,b^{4} d^{2} e^{2} x^{3}+48 A \,a^{2} b^{2} e^{4} x^{2}-288 A a \,b^{3} d \,e^{3} x^{2}+1008 A \,b^{4} d^{2} e^{2} x^{2}-54 B \,a^{3} b \,e^{4} x^{2}+330 B \,a^{2} b^{2} d \,e^{3} x^{2}-1170 B a \,b^{3} d^{2} e^{2} x^{2}+126 B \,b^{4} d^{3} e \,x^{2}-40 A \,a^{3} b \,e^{4} x +216 A \,a^{2} b^{2} d \,e^{3} x -504 A a \,b^{3} d^{2} e^{2} x +840 A \,b^{4} d^{3} e x +45 B \,a^{4} e^{4} x -248 B \,a^{3} b d \,e^{3} x +594 B \,a^{2} b^{2} d^{2} e^{2} x -1008 B a \,b^{3} d^{3} e x +105 B \,b^{4} d^{4} x +35 A \,a^{4} e^{4}-180 A \,a^{3} b d \,e^{3}+378 A \,a^{2} b^{2} d^{2} e^{2}-420 A a \,b^{3} d^{3} e +315 A \,b^{4} d^{4}+10 B \,a^{4} d \,e^{3}-54 B \,a^{3} b \,d^{2} e^{2}+126 B \,a^{2} b^{2} d^{3} e -210 B a \,b^{3} d^{4}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}\right )}\)	\(505\)

Input:

int((B*x+A)/(b*x+a)^(1/2)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

-2/315*(b*x+a)^(1/2)*(128*A*b^4*e^4*x^4-144*B*a*b^3*e^4*x^4+16*B*b^4*d*e^3 
*x^4-64*A*a*b^3*e^4*x^3+576*A*b^4*d*e^3*x^3+72*B*a^2*b^2*e^4*x^3-656*B*a*b 
^3*d*e^3*x^3+72*B*b^4*d^2*e^2*x^3+48*A*a^2*b^2*e^4*x^2-288*A*a*b^3*d*e^3*x 
^2+1008*A*b^4*d^2*e^2*x^2-54*B*a^3*b*e^4*x^2+330*B*a^2*b^2*d*e^3*x^2-1170* 
B*a*b^3*d^2*e^2*x^2+126*B*b^4*d^3*e*x^2-40*A*a^3*b*e^4*x+216*A*a^2*b^2*d*e 
^3*x-504*A*a*b^3*d^2*e^2*x+840*A*b^4*d^3*e*x+45*B*a^4*e^4*x-248*B*a^3*b*d* 
e^3*x+594*B*a^2*b^2*d^2*e^2*x-1008*B*a*b^3*d^3*e*x+105*B*b^4*d^4*x+35*A*a^ 
4*e^4-180*A*a^3*b*d*e^3+378*A*a^2*b^2*d^2*e^2-420*A*a*b^3*d^3*e+315*A*b^4* 
d^4+10*B*a^4*d*e^3-54*B*a^3*b*d^2*e^2+126*B*a^2*b^2*d^3*e-210*B*a*b^3*d^4) 
/(e*x+d)^(9/2)/(a*e-b*d)^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (221) = 442\).

Time = 29.80 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.34 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="fricas")
 

Output:

2/315*(35*A*a^4*e^4 - 105*(2*B*a*b^3 - 3*A*b^4)*d^4 + 42*(3*B*a^2*b^2 - 10 
*A*a*b^3)*d^3*e - 54*(B*a^3*b - 7*A*a^2*b^2)*d^2*e^2 + 10*(B*a^4 - 18*A*a^ 
3*b)*d*e^3 + 16*(B*b^4*d*e^3 - (9*B*a*b^3 - 8*A*b^4)*e^4)*x^4 + 8*(9*B*b^4 
*d^2*e^2 - 2*(41*B*a*b^3 - 36*A*b^4)*d*e^3 + (9*B*a^2*b^2 - 8*A*a*b^3)*e^4 
)*x^3 + 6*(21*B*b^4*d^3*e - 3*(65*B*a*b^3 - 56*A*b^4)*d^2*e^2 + (55*B*a^2* 
b^2 - 48*A*a*b^3)*d*e^3 - (9*B*a^3*b - 8*A*a^2*b^2)*e^4)*x^2 + (105*B*b^4* 
d^4 - 168*(6*B*a*b^3 - 5*A*b^4)*d^3*e + 18*(33*B*a^2*b^2 - 28*A*a*b^3)*d^2 
*e^2 - 8*(31*B*a^3*b - 27*A*a^2*b^2)*d*e^3 + 5*(9*B*a^4 - 8*A*a^3*b)*e^4)* 
x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^5*d^10 - 5*a*b^4*d^9*e + 10*a^2*b^3*d^8* 
e^2 - 10*a^3*b^2*d^7*e^3 + 5*a^4*b*d^6*e^4 - a^5*d^5*e^5 + (b^5*d^5*e^5 - 
5*a*b^4*d^4*e^6 + 10*a^2*b^3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^9 
- a^5*e^10)*x^5 + 5*(b^5*d^6*e^4 - 5*a*b^4*d^5*e^5 + 10*a^2*b^3*d^4*e^6 - 
10*a^3*b^2*d^3*e^7 + 5*a^4*b*d^2*e^8 - a^5*d*e^9)*x^4 + 10*(b^5*d^7*e^3 - 
5*a*b^4*d^6*e^4 + 10*a^2*b^3*d^5*e^5 - 10*a^3*b^2*d^4*e^6 + 5*a^4*b*d^3*e^ 
7 - a^5*d^2*e^8)*x^3 + 10*(b^5*d^8*e^2 - 5*a*b^4*d^7*e^3 + 10*a^2*b^3*d^6* 
e^4 - 10*a^3*b^2*d^5*e^5 + 5*a^4*b*d^4*e^6 - a^5*d^3*e^7)*x^2 + 5*(b^5*d^9 
*e - 5*a*b^4*d^8*e^2 + 10*a^2*b^3*d^7*e^3 - 10*a^3*b^2*d^6*e^4 + 5*a^4*b*d 
^5*e^5 - a^5*d^4*e^6)*x)
 

Sympy [F]

\[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx=\int \frac {A + B x}{\sqrt {a + b x} \left (d + e x\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((B*x+A)/(b*x+a)**(1/2)/(e*x+d)**(11/2),x)
 

Output:

Integral((A + B*x)/(sqrt(a + b*x)*(d + e*x)**(11/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/(b*x+a)^(1/2)/(e*x+d)^(11/2),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(e*(a*e-b*d)>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (221) = 442\).

Time = 0.31 (sec) , antiderivative size = 933, normalized size of antiderivative = 3.72 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="giac")
 

Output:

2/315*((2*(4*(b*x + a)*(2*(B*b^10*d*e^7*abs(b) - 9*B*a*b^9*e^8*abs(b) + 8* 
A*b^10*e^8*abs(b))*(b*x + a)/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5 + 10*a^2*b^5*d 
^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9) + 9*(B*b^11*d 
^2*e^6*abs(b) - 10*B*a*b^10*d*e^7*abs(b) + 8*A*b^11*d*e^7*abs(b) + 9*B*a^2 
*b^9*e^8*abs(b) - 8*A*a*b^10*e^8*abs(b))/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5 + 
10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9)) 
+ 63*(B*b^12*d^3*e^5*abs(b) - 11*B*a*b^11*d^2*e^6*abs(b) + 8*A*b^12*d^2*e^ 
6*abs(b) + 19*B*a^2*b^10*d*e^7*abs(b) - 16*A*a*b^11*d*e^7*abs(b) - 9*B*a^3 
*b^9*e^8*abs(b) + 8*A*a^2*b^10*e^8*abs(b))/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5 
+ 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9) 
)*(b*x + a) + 105*(B*b^13*d^4*e^4*abs(b) - 12*B*a*b^12*d^3*e^5*abs(b) + 8* 
A*b^13*d^3*e^5*abs(b) + 30*B*a^2*b^11*d^2*e^6*abs(b) - 24*A*a*b^12*d^2*e^6 
*abs(b) - 28*B*a^3*b^10*d*e^7*abs(b) + 24*A*a^2*b^11*d*e^7*abs(b) + 9*B*a^ 
4*b^9*e^8*abs(b) - 8*A*a^3*b^10*e^8*abs(b))/(b^7*d^5*e^4 - 5*a*b^6*d^4*e^5 
 + 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*a^4*b^3*d*e^8 - a^5*b^2*e^9 
))*(b*x + a) - 315*(B*a*b^13*d^4*e^4*abs(b) - A*b^14*d^4*e^4*abs(b) - 4*B* 
a^2*b^12*d^3*e^5*abs(b) + 4*A*a*b^13*d^3*e^5*abs(b) + 6*B*a^3*b^11*d^2*e^6 
*abs(b) - 6*A*a^2*b^12*d^2*e^6*abs(b) - 4*B*a^4*b^10*d*e^7*abs(b) + 4*A*a^ 
3*b^11*d*e^7*abs(b) + B*a^5*b^9*e^8*abs(b) - A*a^4*b^10*e^8*abs(b))/(b^7*d 
^5*e^4 - 5*a*b^6*d^4*e^5 + 10*a^2*b^5*d^3*e^6 - 10*a^3*b^4*d^2*e^7 + 5*...
 

Mupad [B] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.27 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {20\,B\,a^5\,d\,e^3+70\,A\,a^5\,e^4-108\,B\,a^4\,b\,d^2\,e^2-360\,A\,a^4\,b\,d\,e^3+252\,B\,a^3\,b^2\,d^3\,e+756\,A\,a^3\,b^2\,d^2\,e^2-420\,B\,a^2\,b^3\,d^4-840\,A\,a^2\,b^3\,d^3\,e+630\,A\,a\,b^4\,d^4}{315\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\left (90\,B\,a^5\,e^4-476\,B\,a^4\,b\,d\,e^3-10\,A\,a^4\,b\,e^4+1080\,B\,a^3\,b^2\,d^2\,e^2+72\,A\,a^3\,b^2\,d\,e^3-1764\,B\,a^2\,b^3\,d^3\,e-252\,A\,a^2\,b^3\,d^2\,e^2-210\,B\,a\,b^4\,d^4+840\,A\,a\,b^4\,d^3\,e+630\,A\,b^5\,d^4\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^4\,x^5\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )}{315\,e^2\,{\left (a\,e-b\,d\right )}^5}+\frac {16\,b^3\,x^4\,\left (a\,e+9\,b\,d\right )\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )}{315\,e^3\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^2\,x^3\,\left (-a^2\,e^2+18\,a\,b\,d\,e+63\,b^2\,d^2\right )\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^5}+\frac {2\,b\,x^2\,\left (8\,A\,b\,e-9\,B\,a\,e+B\,b\,d\right )\,\left (a^3\,e^3-9\,a^2\,b\,d\,e^2+63\,a\,b^2\,d^2\,e+105\,b^3\,d^3\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^5}\right )}{x^5\,\sqrt {a+b\,x}+\frac {d^5\,\sqrt {a+b\,x}}{e^5}+\frac {10\,d^2\,x^3\,\sqrt {a+b\,x}}{e^2}+\frac {10\,d^3\,x^2\,\sqrt {a+b\,x}}{e^3}+\frac {5\,d\,x^4\,\sqrt {a+b\,x}}{e}+\frac {5\,d^4\,x\,\sqrt {a+b\,x}}{e^4}} \] Input:

int((A + B*x)/((a + b*x)^(1/2)*(d + e*x)^(11/2)),x)
 

Output:

-((d + e*x)^(1/2)*((70*A*a^5*e^4 + 630*A*a*b^4*d^4 + 20*B*a^5*d*e^3 - 420* 
B*a^2*b^3*d^4 - 840*A*a^2*b^3*d^3*e + 252*B*a^3*b^2*d^3*e - 108*B*a^4*b*d^ 
2*e^2 + 756*A*a^3*b^2*d^2*e^2 - 360*A*a^4*b*d*e^3)/(315*e^5*(a*e - b*d)^5) 
 + (x*(630*A*b^5*d^4 + 90*B*a^5*e^4 - 10*A*a^4*b*e^4 - 210*B*a*b^4*d^4 + 7 
2*A*a^3*b^2*d*e^3 - 1764*B*a^2*b^3*d^3*e - 252*A*a^2*b^3*d^2*e^2 + 1080*B* 
a^3*b^2*d^2*e^2 + 840*A*a*b^4*d^3*e - 476*B*a^4*b*d*e^3))/(315*e^5*(a*e - 
b*d)^5) + (32*b^4*x^5*(8*A*b*e - 9*B*a*e + B*b*d))/(315*e^2*(a*e - b*d)^5) 
 + (16*b^3*x^4*(a*e + 9*b*d)*(8*A*b*e - 9*B*a*e + B*b*d))/(315*e^3*(a*e - 
b*d)^5) + (4*b^2*x^3*(63*b^2*d^2 - a^2*e^2 + 18*a*b*d*e)*(8*A*b*e - 9*B*a* 
e + B*b*d))/(315*e^4*(a*e - b*d)^5) + (2*b*x^2*(8*A*b*e - 9*B*a*e + B*b*d) 
*(a^3*e^3 + 105*b^3*d^3 + 63*a*b^2*d^2*e - 9*a^2*b*d*e^2))/(315*e^5*(a*e - 
 b*d)^5)))/(x^5*(a + b*x)^(1/2) + (d^5*(a + b*x)^(1/2))/e^5 + (10*d^2*x^3* 
(a + b*x)^(1/2))/e^2 + (10*d^3*x^2*(a + b*x)^(1/2))/e^3 + (5*d*x^4*(a + b* 
x)^(1/2))/e + (5*d^4*x*(a + b*x)^(1/2))/e^4)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 845, normalized size of antiderivative = 3.37 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{11/2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(b*x+a)^(1/2)/(e*x+d)^(11/2),x)
 

Output:

(2*( - 35*sqrt(d + e*x)*sqrt(a + b*x)*a**4*e**5 + 135*sqrt(d + e*x)*sqrt(a 
 + b*x)*a**3*b*d*e**4 - 5*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b*e**5*x - 189* 
sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**2*d**2*e**3 + 27*sqrt(d + e*x)*sqrt(a 
+ b*x)*a**2*b**2*d*e**4*x + 6*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**2*e**5*x 
**2 + 105*sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*d**3*e**2 - 63*sqrt(d + e*x)* 
sqrt(a + b*x)*a*b**3*d**2*e**3*x - 36*sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*d 
*e**4*x**2 - 8*sqrt(d + e*x)*sqrt(a + b*x)*a*b**3*e**5*x**3 + 105*sqrt(d + 
 e*x)*sqrt(a + b*x)*b**4*d**3*e**2*x + 126*sqrt(d + e*x)*sqrt(a + b*x)*b** 
4*d**2*e**3*x**2 + 72*sqrt(d + e*x)*sqrt(a + b*x)*b**4*d*e**4*x**3 + 16*sq 
rt(d + e*x)*sqrt(a + b*x)*b**4*e**5*x**4 - 16*sqrt(e)*sqrt(b)*b**4*d**5 - 
80*sqrt(e)*sqrt(b)*b**4*d**4*e*x - 160*sqrt(e)*sqrt(b)*b**4*d**3*e**2*x**2 
 - 160*sqrt(e)*sqrt(b)*b**4*d**2*e**3*x**3 - 80*sqrt(e)*sqrt(b)*b**4*d*e** 
4*x**4 - 16*sqrt(e)*sqrt(b)*b**4*e**5*x**5))/(315*e**2*(a**4*d**5*e**4 + 5 
*a**4*d**4*e**5*x + 10*a**4*d**3*e**6*x**2 + 10*a**4*d**2*e**7*x**3 + 5*a* 
*4*d*e**8*x**4 + a**4*e**9*x**5 - 4*a**3*b*d**6*e**3 - 20*a**3*b*d**5*e**4 
*x - 40*a**3*b*d**4*e**5*x**2 - 40*a**3*b*d**3*e**6*x**3 - 20*a**3*b*d**2* 
e**7*x**4 - 4*a**3*b*d*e**8*x**5 + 6*a**2*b**2*d**7*e**2 + 30*a**2*b**2*d* 
*6*e**3*x + 60*a**2*b**2*d**5*e**4*x**2 + 60*a**2*b**2*d**4*e**5*x**3 + 30 
*a**2*b**2*d**3*e**6*x**4 + 6*a**2*b**2*d**2*e**7*x**5 - 4*a*b**3*d**8*e - 
 20*a*b**3*d**7*e**2*x - 40*a*b**3*d**6*e**3*x**2 - 40*a*b**3*d**5*e**4...