\(\int x \sqrt {a+b x} (c+d x)^{5/2} \, dx\) [183]

Optimal result

Integrand size = 20, antiderivative size = 268 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=-\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {(b c-a d)^4 (3 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}} \] Output:

-1/128*(-a*d+b*c)^3*(7*a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^2-1/64 
*(-a*d+b*c)^2*(7*a*d+3*b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^4/d-1/48*(-a*d+b 
*c)*(7*a*d+3*b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b^3/d-1/40*(7*a*d+3*b*c)*(b* 
x+a)^(3/2)*(d*x+c)^(5/2)/b^2/d+1/5*(b*x+a)^(3/2)*(d*x+c)^(7/2)/b/d+1/128*( 
-a*d+b*c)^4*(7*a*d+3*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1 
/2))/b^(9/2)/d^(5/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.86 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (34 c+7 d x)-2 a^2 b^2 d^2 \left (173 c^2+111 c d x+28 d^2 x^2\right )+2 a b^3 d \left (30 c^3+109 c^2 d x+88 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^4 d^2}+\frac {(b c-a d)^4 (3 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}} \] Input:

Integrate[x*Sqrt[a + b*x]*(c + d*x)^(5/2),x]
 

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(34*c + 7*d*x) - 
 2*a^2*b^2*d^2*(173*c^2 + 111*c*d*x + 28*d^2*x^2) + 2*a*b^3*d*(30*c^3 + 10 
9*c^2*d*x + 88*c*d^2*x^2 + 24*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x + 744*c 
^2*d^2*x^2 + 1008*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^4*d^2) + ((b*c - a*d) 
^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x]) 
])/(128*b^(9/2)*d^(5/2))
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {90, 60, 60, 60, 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.

\(\displaystyle \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \int \sqrt {a+b x} (c+d x)^{5/2}dx}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \left (\frac {5 (b c-a d) \int \sqrt {a+b x} (c+d x)^{3/2}dx}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\right )}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \left (\frac {5 (b c-a d) \left (\frac {(b c-a d) \int \sqrt {a+b x} \sqrt {c+d x}dx}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\right )}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \left (\frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\right )}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \left (\frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\right )}{10 b d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \left (\frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\right )}{10 b d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(7 a d+3 b c) \left (\frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\right )}{10 b d}\)

Input:

Int[x*Sqrt[a + b*x]*(c + d*x)^(5/2),x]
 

Output:

((a + b*x)^(3/2)*(c + d*x)^(7/2))/(5*b*d) - ((3*b*c + 7*a*d)*(((a + b*x)^( 
3/2)*(c + d*x)^(5/2))/(4*b) + (5*(b*c - a*d)*(((a + b*x)^(3/2)*(c + d*x)^( 
3/2))/(3*b) + ((b*c - a*d)*(((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*b) + ((b*c 
- a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sq 
rt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(4*b)))/(2*b))) 
/(8*b)))/(10*b*d)
 

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 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 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 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. \(787\) vs. \(2(224)=448\).

Time = 0.23 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.94

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+96 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+2016 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-112 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+352 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+1488 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{5} d^{5}-375 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{4} b c \,d^{4}+450 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} b^{2} c^{2} d^{3}-150 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b^{3} c^{3} d^{2}-75 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{4} c^{4} d +45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{5} c^{5}+140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} b \,d^{4} x -444 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b^{2} c \,d^{3} x +436 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{3} c^{2} d^{2} x +60 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{4} c^{3} d x -210 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{4} d^{4}+680 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} b c \,d^{3}-692 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b^{2} c^{2} d^{2}+120 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{3} c^{3} d -90 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{4} c^{4}\right )}{3840 d^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{4} \sqrt {d b}}\)	\(788\)

Input:

int(x*(b*x+a)^(1/2)*(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)+96*a*b^3*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+2016*b^ 
4*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-112*a^2*b^2*d^4*x^2*((b*x+ 
a)*(d*x+c))^(1/2)*(d*b)^(1/2)+352*a*b^3*c*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)* 
(d*b)^(1/2)+1488*b^4*c^2*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+105*l 
n(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^5*d^5-375*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^4*b*c*d^4+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^2*d^3-150*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^3*d^2-75* 
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^4*d+45*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))*b^5*c^5+140*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b*d 
^4*x-444*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^2*c*d^3*x+436*((b*x+a)* 
(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3*c^2*d^2*x+60*((b*x+a)*(d*x+c))^(1/2)*(d*b 
)^(1/2)*b^4*c^3*d*x-210*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4*d^4+680*(( 
b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b*c*d^3-692*((b*x+a)*(d*x+c))^(1/2)* 
(d*b)^(1/2)*a^2*b^2*c^2*d^2+120*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3* 
c^3*d-90*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^4*c^4)/d^2/((b*x+a)*(d*x+c) 
)^(1/2)/b^4/(d*b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.63 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\left [\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{5} d^{3}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{5} d^{3}}\right ] \] Input:

integrate(x*(b*x+a)^(1/2)*(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/7680*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c 
^2*d^3 - 25*a^4*b*c*d^4 + 7*a^5*d^5)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 
 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*s 
qrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(384*b^5*d^5*x^4 - 45*b^5*c^4* 
d + 60*a*b^4*c^3*d^2 - 346*a^2*b^3*c^2*d^3 + 340*a^3*b^2*c*d^4 - 105*a^4*b 
*d^5 + 48*(21*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 22*a*b^4*c* 
d^4 - 7*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 109*a*b^4*c^2*d^3 - 111*a^2 
*b^3*c*d^4 + 35*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^3), -1 
/3840*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2 
*d^3 - 25*a^4*b*c*d^4 + 7*a^5*d^5)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + 
a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2* 
c*d + a*b*d^2)*x)) - 2*(384*b^5*d^5*x^4 - 45*b^5*c^4*d + 60*a*b^4*c^3*d^2 
- 346*a^2*b^3*c^2*d^3 + 340*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(21*b^5*c*d 
^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 22*a*b^4*c*d^4 - 7*a^2*b^3*d^5)* 
x^2 + 2*(15*b^5*c^3*d^2 + 109*a*b^4*c^2*d^3 - 111*a^2*b^3*c*d^4 + 35*a^3*b 
^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^3)]
 

Sympy [F]

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

integrate(x*(b*x+a)**(1/2)*(d*x+c)**(5/2),x)
 

Output:

Integral(x*sqrt(a + b*x)*(c + d*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x*(b*x+a)^(1/2)*(d*x+c)^(5/2),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. 1449 vs. \(2 (224) = 448\).

Time = 0.30 (sec) , antiderivative size = 1449, normalized size of antiderivative = 5.41 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x+a)^(1/2)*(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

1/1920*(20*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)* 
(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2* 
d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 
+ 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt 
(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c* 
d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a 
)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*c*d*abs(b)/b + 10*(sqrt(b^2*c + (b*x 
 + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^ 
5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a 
^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b 
^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4* 
a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sq 
rt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b 
^2*d^3))*a*d^2*abs(b)/b^2 + 480*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b* 
x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 
3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d 
 - a*b*d)))/(sqrt(b*d)*d))*a*c^2*abs(b)/b^3 + 80*(sqrt(b^2*c + (b*x + a)*b 
*d - a*b*d)*(2*(4*b*x + 4*a + (b*c*d^3 - 13*a*d^4)/d^4)*(b*x + a) - 3*(b^2 
*c^2*d^2 + 2*a*b*c*d^3 - 11*a^2*d^4)/d^4)*sqrt(b*x + a) - 3*(b^4*c^3 + a*b 
^3*c^2*d + 3*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x +...
 

Mupad [F(-1)]

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

int(x*(a + b*x)^(1/2)*(c + d*x)^(5/2),x)
 

Output:

int(x*(a + b*x)^(1/2)*(c + d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.43 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {-105 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{4} b \,d^{5}+340 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} c \,d^{4}+70 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} d^{5} x -346 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2} d^{3}-222 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c \,d^{4} x -56 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} d^{5} x^{2}+60 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{3} d^{2}+218 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{2} d^{3} x +176 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c \,d^{4} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} d^{5} x^{3}-45 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{4} d +30 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{3} d^{2} x +744 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{2} d^{3} x^{2}+1008 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c \,d^{4} x^{3}+384 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} d^{5} x^{4}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{5} d^{5}-375 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} b c \,d^{4}+450 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b^{2} c^{2} d^{3}-150 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{3} c^{3} d^{2}-75 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{4} c^{4} d +45 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{5} c^{5}}{1920 b^{5} d^{3}} \] Input:

int(x*(b*x+a)^(1/2)*(d*x+c)^(5/2),x)
 

Output:

( - 105*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*d**5 + 340*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**3*b**2*c*d**4 + 70*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*d**5*x 
 - 346*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c**2*d**3 - 222*sqrt(c + d*x) 
*sqrt(a + b*x)*a**2*b**3*c*d**4*x - 56*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b* 
*3*d**5*x**2 + 60*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**3*d**2 + 218*sqrt( 
c + d*x)*sqrt(a + b*x)*a*b**4*c**2*d**3*x + 176*sqrt(c + d*x)*sqrt(a + b*x 
)*a*b**4*c*d**4*x**2 + 48*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*d**5*x**3 - 4 
5*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**4*d + 30*sqrt(c + d*x)*sqrt(a + b*x) 
*b**5*c**3*d**2*x + 744*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**2*d**3*x**2 + 
1008*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c*d**4*x**3 + 384*sqrt(c + d*x)*sqrt 
(a + b*x)*b**5*d**5*x**4 + 105*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) 
+ sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**5*d**5 - 375*sqrt(d)*sqrt(b)* 
log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4* 
b*c*d**4 + 450*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
 + d*x))/sqrt(a*d - b*c))*a**3*b**2*c**2*d**3 - 150*sqrt(d)*sqrt(b)*log((s 
qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**3*c 
**3*d**2 - 75*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
+ d*x))/sqrt(a*d - b*c))*a*b**4*c**4*d + 45*sqrt(d)*sqrt(b)*log((sqrt(d)*s 
qrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**5*c**5)/(1920*b* 
*5*d**3)