\(\int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx\) [306]

Optimal result

Integrand size = 20, antiderivative size = 220 \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=-\frac {a^4 (12 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{512 b^3}+\frac {a^3 (12 A b-5 a B) x^{3/2} \sqrt {a+b x}}{768 b^2}+\frac {31 a^2 (12 A b-5 a B) x^{5/2} \sqrt {a+b x}}{960 b}+\frac {7}{160} a (12 A b-5 a B) x^{7/2} \sqrt {a+b x}+\frac {1}{60} b (12 A b-5 a B) x^{9/2} \sqrt {a+b x}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}+\frac {a^5 (12 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{7/2}} \] Output:

-1/512*a^4*(12*A*b-5*B*a)*x^(1/2)*(b*x+a)^(1/2)/b^3+1/768*a^3*(12*A*b-5*B* 
a)*x^(3/2)*(b*x+a)^(1/2)/b^2+31/960*a^2*(12*A*b-5*B*a)*x^(5/2)*(b*x+a)^(1/ 
2)/b+7/160*a*(12*A*b-5*B*a)*x^(7/2)*(b*x+a)^(1/2)+1/60*b*(12*A*b-5*B*a)*x^ 
(9/2)*(b*x+a)^(1/2)+1/6*B*x^(5/2)*(b*x+a)^(7/2)/b+1/512*a^5*(12*A*b-5*B*a) 
*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.75 \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (75 a^5 B+40 a^3 b^2 x (3 A+B x)+256 b^5 x^4 (6 A+5 B x)-10 a^4 b (18 A+5 B x)+48 a^2 b^3 x^2 (62 A+45 B x)+64 a b^4 x^3 (63 A+50 B x)\right )+30 a^5 (-12 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )}{7680 b^{7/2}} \] Input:

Integrate[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]
 

Output:

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(75*a^5*B + 40*a^3*b^2*x*(3*A + B*x) + 256* 
b^5*x^4*(6*A + 5*B*x) - 10*a^4*b*(18*A + 5*B*x) + 48*a^2*b^3*x^2*(62*A + 4 
5*B*x) + 64*a*b^4*x^3*(63*A + 50*B*x)) + 30*a^5*(-12*A*b + 5*a*B)*ArcTanh[ 
(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x])])/(7680*b^(7/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {90, 60, 60, 60, 60, 60, 65, 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.

\(\displaystyle \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(12 A b-5 a B) \int x^{3/2} (a+b x)^{5/2}dx}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \int x^{3/2} (a+b x)^{3/2}dx+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \left (\frac {3}{8} a \int x^{3/2} \sqrt {a+b x}dx+\frac {1}{4} x^{5/2} (a+b x)^{3/2}\right )+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \int \frac {x^{3/2}}{\sqrt {a+b x}}dx+\frac {1}{3} x^{5/2} \sqrt {a+b x}\right )+\frac {1}{4} x^{5/2} (a+b x)^{3/2}\right )+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a+b x}}dx}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\right )+\frac {1}{4} x^{5/2} (a+b x)^{3/2}\right )+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{2 b}\right )}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\right )+\frac {1}{4} x^{5/2} (a+b x)^{3/2}\right )+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}\right )}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\right )+\frac {1}{4} x^{5/2} (a+b x)^{3/2}\right )+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(12 A b-5 a B) \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\right )}{4 b}\right )+\frac {1}{3} x^{5/2} \sqrt {a+b x}\right )+\frac {1}{4} x^{5/2} (a+b x)^{3/2}\right )+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\right )}{12 b}+\frac {B x^{5/2} (a+b x)^{7/2}}{6 b}\)

Input:

Int[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]
 

Output:

(B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + ((12*A*b - 5*a*B)*((x^(5/2)*(a + b*x)^ 
(5/2))/5 + (a*((x^(5/2)*(a + b*x)^(3/2))/4 + (3*a*((x^(5/2)*Sqrt[a + b*x]) 
/3 + (a*((x^(3/2)*Sqrt[a + b*x])/(2*b) - (3*a*((Sqrt[x]*Sqrt[a + b*x])/b - 
 (a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/2)))/(4*b)))/6))/8))/2) 
)/(12*b)
 

Defintions of rubi rules used

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 65

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

rule 90

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

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])
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {\left (-1280 b^{5} B \,x^{5}-1536 A \,b^{5} x^{4}-3200 B a \,b^{4} x^{4}-4032 A a \,b^{4} x^{3}-2160 B \,a^{2} b^{3} x^{3}-2976 A \,a^{2} b^{3} x^{2}-40 B \,a^{3} b^{2} x^{2}-120 a^{3} b^{2} A x +50 a^{4} b B x +180 a^{4} b A -75 a^{5} B \right ) \sqrt {x}\, \sqrt {b x +a}}{7680 b^{3}}+\frac {a^{5} \left (12 A b -5 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {b x +a}}\)	\(183\)
default	\(\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (2560 B \,b^{\frac {11}{2}} x^{5} \sqrt {x \left (b x +a \right )}+3072 A \,b^{\frac {11}{2}} x^{4} \sqrt {x \left (b x +a \right )}+6400 B a \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}+8064 A a \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}+4320 B \,a^{2} b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}+5952 A \,a^{2} b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+80 B \,a^{3} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+240 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a^{3} x -100 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{4} x +180 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5} b -360 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{4}-75 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{6}+150 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{5}\right )}{15360 b^{\frac {7}{2}} \sqrt {x \left (b x +a \right )}}\)	\(302\)

Input:

int(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x,method=_RETURNVERBOSE)
 

Output:

-1/7680/b^3*(-1280*B*b^5*x^5-1536*A*b^5*x^4-3200*B*a*b^4*x^4-4032*A*a*b^4* 
x^3-2160*B*a^2*b^3*x^3-2976*A*a^2*b^3*x^2-40*B*a^3*b^2*x^2-120*A*a^3*b^2*x 
+50*B*a^4*b*x+180*A*a^4*b-75*B*a^5)*x^(1/2)*(b*x+a)^(1/2)+1/1024*a^5/b^(7/ 
2)*(12*A*b-5*B*a)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))*(x*(b*x+a))^(1 
/2)/x^(1/2)/(b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.55 \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\left [-\frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (1280 \, B b^{6} x^{5} + 75 \, B a^{5} b - 180 \, A a^{4} b^{2} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{3} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{2} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15360 \, b^{4}}, \frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) + {\left (1280 \, B b^{6} x^{5} + 75 \, B a^{5} b - 180 \, A a^{4} b^{2} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{3} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{2} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{7680 \, b^{4}}\right ] \] Input:

integrate(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x, algorithm="fricas")
 

Output:

[-1/15360*(15*(5*B*a^6 - 12*A*a^5*b)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*s 
qrt(b)*sqrt(x) + a) - 2*(1280*B*b^6*x^5 + 75*B*a^5*b - 180*A*a^4*b^2 + 128 
*(25*B*a*b^5 + 12*A*b^6)*x^4 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^3 + 8*(5* 
B*a^3*b^3 + 372*A*a^2*b^4)*x^2 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x)*sqrt(b 
*x + a)*sqrt(x))/b^4, 1/7680*(15*(5*B*a^6 - 12*A*a^5*b)*sqrt(-b)*arctan(sq 
rt(-b)*sqrt(x)/sqrt(b*x + a)) + (1280*B*b^6*x^5 + 75*B*a^5*b - 180*A*a^4*b 
^2 + 128*(25*B*a*b^5 + 12*A*b^6)*x^4 + 144*(15*B*a^2*b^4 + 28*A*a*b^5)*x^3 
 + 8*(5*B*a^3*b^3 + 372*A*a^2*b^4)*x^2 - 10*(5*B*a^4*b^2 - 12*A*a^3*b^3)*x 
)*sqrt(b*x + a)*sqrt(x))/b^4]
 

Sympy [F(-1)]

Timed out. \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\text {Timed out} \] Input:

integrate(x**(3/2)*(b*x+a)**(5/2)*(B*x+A),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (174) = 348\).

Time = 0.05 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.92 \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\frac {1}{6} \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B x + \frac {1}{4} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a x - \frac {7 \, \sqrt {b x^{2} + a x} B a^{4} x}{256 \, b^{2}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2} x}{96 \, b} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{3} x}{32 \, b} + \frac {7 \, B a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {5}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{5}}{512 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{3}}{192 \, b^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{4}}{64 \, b^{2}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{60 \, b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a^{2}}{8 \, b} + \frac {3 \, \sqrt {b x^{2} + a x} {\left (B a + A b\right )} a^{3} x}{64 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (B a + A b\right )} a x}{8 \, b} - \frac {3 \, {\left (B a + A b\right )} a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} + \frac {3 \, \sqrt {b x^{2} + a x} {\left (B a + A b\right )} a^{4}}{128 \, b^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (B a + A b\right )} a^{2}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} {\left (B a + A b\right )}}{5 \, b} \] Input:

integrate(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x, algorithm="maxima")
 

Output:

1/6*(b*x^2 + a*x)^(5/2)*B*x + 1/4*(b*x^2 + a*x)^(3/2)*A*a*x - 7/256*sqrt(b 
*x^2 + a*x)*B*a^4*x/b^2 + 7/96*(b*x^2 + a*x)^(3/2)*B*a^2*x/b - 3/32*sqrt(b 
*x^2 + a*x)*A*a^3*x/b + 7/1024*B*a^6*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*s 
qrt(b))/b^(7/2) + 3/128*A*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) 
/b^(5/2) - 7/512*sqrt(b*x^2 + a*x)*B*a^5/b^3 + 7/192*(b*x^2 + a*x)^(3/2)*B 
*a^3/b^2 - 3/64*sqrt(b*x^2 + a*x)*A*a^4/b^2 - 7/60*(b*x^2 + a*x)^(5/2)*B*a 
/b + 1/8*(b*x^2 + a*x)^(3/2)*A*a^2/b + 3/64*sqrt(b*x^2 + a*x)*(B*a + A*b)* 
a^3*x/b^2 - 1/8*(b*x^2 + a*x)^(3/2)*(B*a + A*b)*a*x/b - 3/256*(B*a + A*b)* 
a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 3/128*sqrt(b*x^ 
2 + a*x)*(B*a + A*b)*a^4/b^3 - 1/16*(b*x^2 + a*x)^(3/2)*(B*a + A*b)*a^2/b^ 
2 + 1/5*(b*x^2 + a*x)^(5/2)*(B*a + A*b)/b
 

Giac [F(-1)]

Timed out. \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\text {Timed out} \] Input:

integrate(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\int x^{3/2}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2} \,d x \] Input:

int(x^(3/2)*(A + B*x)*(a + b*x)^(5/2),x)
 

Output:

int(x^(3/2)*(A + B*x)*(a + b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.60 \[ \int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx=\frac {-105 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b +70 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} x +3016 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} x^{2}+6192 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} x^{3}+4736 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} x^{4}+1280 \sqrt {x}\, \sqrt {b x +a}\, b^{6} x^{5}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6}}{7680 b^{3}} \] Input:

int(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x)
 

Output:

( - 105*sqrt(x)*sqrt(a + b*x)*a**5*b + 70*sqrt(x)*sqrt(a + b*x)*a**4*b**2* 
x + 3016*sqrt(x)*sqrt(a + b*x)*a**3*b**3*x**2 + 6192*sqrt(x)*sqrt(a + b*x) 
*a**2*b**4*x**3 + 4736*sqrt(x)*sqrt(a + b*x)*a*b**5*x**4 + 1280*sqrt(x)*sq 
rt(a + b*x)*b**6*x**5 + 105*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/ 
sqrt(a))*a**6)/(7680*b**3)