\(\int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx\) [214]

Optimal result

Integrand size = 22, antiderivative size = 314 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=-\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+13 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b^2 d}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}} \] Output:

-1/128*(-a*d+b*c)^2*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^ 
(1/2)/b^3/d^4+1/192*(-a*d+b*c)*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*(b*x+a)^(3/ 
2)*(d*x+c)^(1/2)/b^3/d^3+1/48*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*(b*x+a)^(5/2 
)*(d*x+c)^(1/2)/b^3/d^2-1/40*(13*a*d+7*b*c)*(b*x+a)^(5/2)*(d*x+c)^(3/2)/b^ 
2/d^2+1/5*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2/d+1/128*(-a*d+b*c)^3*(3*a^2*d^2+ 
6*a*b*c*d+7*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/ 
b^(7/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (45 a^4 d^4-30 a^3 b d^3 (c+d x)+6 a^2 b^2 d^2 \left (-6 c^2+3 c d x+4 d^2 x^2\right )+2 a b^3 d \left (95 c^3-61 c^2 d x+48 c d^2 x^2+264 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^3 d^4}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{7/2} d^{9/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(45*a^4*d^4 - 30*a^3*b*d^3*(c + d*x) + 6*a^2* 
b^2*d^2*(-6*c^2 + 3*c*d*x + 4*d^2*x^2) + 2*a*b^3*d*(95*c^3 - 61*c^2*d*x + 
48*c*d^2*x^2 + 264*d^3*x^3) + b^4*(-105*c^4 + 70*c^3*d*x - 56*c^2*d^2*x^2 
+ 48*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^3*d^4) + ((b*c - a*d)^3*(7*b^2*c^2 
 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a 
+ b*x])])/(128*b^(7/2)*d^(9/2))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 27, 90, 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.

\(\displaystyle \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx\)

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{4 b d}-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \int (a+b x)^{3/2} \sqrt {c+d x}dx}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{4 b d}-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac {(b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{4 b d}-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{4 b d}-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{4 b d}-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b d}}{10 b d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{4 b d}-\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b d}}{10 b d}\)

Input:

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

Output:

(x*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(5*b*d) - (((7*b*c + 5*a*d)*(a + b*x)^ 
(5/2)*(c + d*x)^(3/2))/(4*b*d) - (5*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*(( 
(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*b) + ((b*c - a*d)*(((a + b*x)^(3/2)*Sqrt 
[c + d*x])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c 
 - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b] 
*d^(3/2))))/(4*d)))/(6*b)))/(8*b*d))/(10*b*d)
 

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 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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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. \(2(270)=540\).

Time = 0.22 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.51

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}-1056 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-96 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-48 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-192 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+112 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+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 ) a^{5} d^{5}-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 ) a^{4} b c \,d^{4}-30 \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}-90 \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}+225 \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 -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 ) b^{5} c^{5}+60 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} b \,d^{4} x -36 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b^{2} c \,d^{3} x +244 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{3} c^{2} d^{2} x -140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{4} c^{3} d x -90 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{4} d^{4}+60 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{3} b c \,d^{3}+72 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b^{2} c^{2} d^{2}-380 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{3} c^{3} d +210 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{4} c^{4}\right )}{3840 b^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, d^{4} \sqrt {d b}}\)	\(788\)

Input:

int(x^2*(b*x+a)^(3/2)*(d*x+c)^(1/2),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)-1056*a*b^3*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-96* 
b^4*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-48*a^2*b^2*d^4*x^2*((b*x 
+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-192*a*b^3*c*d^3*x^2*((b*x+a)*(d*x+c))^(1/2) 
*(d*b)^(1/2)+112*b^4*c^2*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+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))* 
a^5*d^5-45*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))*a^4*b*c*d^4-30*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^3-90*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+225*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-105*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+60*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b*d^4* 
x-36*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^2*c*d^3*x+244*((b*x+a)*(d*x 
+c))^(1/2)*(d*b)^(1/2)*a*b^3*c^2*d^2*x-140*((b*x+a)*(d*x+c))^(1/2)*(d*b)^( 
1/2)*b^4*c^3*d*x-90*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4*d^4+60*((b*x+a 
)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b*c*d^3+72*((b*x+a)*(d*x+c))^(1/2)*(d*b)^ 
(1/2)*a^2*b^2*c^2*d^2-380*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^3*c^3*d+ 
210*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^4*c^4)/b^3/((b*x+a)*(d*x+c))^(1/ 
2)/d^4/(d*b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.24 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\left [-\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 190 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} - 30 \, a^{3} b^{2} c d^{4} + 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 12 \, a b^{4} c d^{4} - 3 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 61 \, a b^{4} c^{2} d^{3} + 9 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{4} d^{5}}, -\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 190 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} - 30 \, a^{3} b^{2} c d^{4} + 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 12 \, a b^{4} c d^{4} - 3 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 61 \, a b^{4} c^{2} d^{3} + 9 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{4} d^{5}}\right ] \] Input:

integrate(x^2*(b*x+a)^(3/2)*(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

[-1/7680*(15*(7*b^5*c^5 - 15*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c 
^2*d^3 + 3*a^4*b*c*d^4 - 3*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)*sq 
rt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(384*b^5*d^5*x^4 - 105*b^5*c^4* 
d + 190*a*b^4*c^3*d^2 - 36*a^2*b^3*c^2*d^3 - 30*a^3*b^2*c*d^4 + 45*a^4*b*d 
^5 + 48*(b^5*c*d^4 + 11*a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 12*a*b^4*c*d^4 
 - 3*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 61*a*b^4*c^2*d^3 + 9*a^2*b^3*c 
*d^4 - 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^5), -1/3840* 
(15*(7*b^5*c^5 - 15*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 + 
3*a^4*b*c*d^4 - 3*a^5*d^5)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqr 
t(-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 - 105*b^5*c^4*d + 190*a*b^4*c^3*d^2 - 36*a 
^2*b^3*c^2*d^3 - 30*a^3*b^2*c*d^4 + 45*a^4*b*d^5 + 48*(b^5*c*d^4 + 11*a*b^ 
4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 12*a*b^4*c*d^4 - 3*a^2*b^3*d^5)*x^2 + 2*(3 
5*b^5*c^3*d^2 - 61*a*b^4*c^2*d^3 + 9*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x)*sq 
rt(b*x + a)*sqrt(d*x + c))/(b^4*d^5)]
 

Sympy [F]

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

integrate(x**2*(b*x+a)**(3/2)*(d*x+c)**(1/2),x)
 

Output:

Integral(x**2*(a + b*x)**(3/2)*sqrt(c + d*x), x)
 

Maxima [F(-2)]

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

integrate(x^2*(b*x+a)^(3/2)*(d*x+c)^(1/2),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. 827 vs. \(2 (270) = 540\).

Time = 0.22 (sec) , antiderivative size = 827, normalized size of antiderivative = 2.63 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx =\text {Too large to display} \] Input:

integrate(x^2*(b*x+a)^(3/2)*(d*x+c)^(1/2),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))*a*abs(b)/b + 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 
 + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^2))*a^2*abs(b)/ 
b^4 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(6*(8*b*x + 8*a + (b*c*d^ 
7 - 41*a*d^8)/d^8)*(b*x + a) - (7*b^2*c^2*d^6 + 26*a*b*c*d^7 - 513*a^2*d^8 
)/d^8)*(b*x + a) + 5*(7*b^3*c^3*d^5 + 19*a*b^2*c^2*d^6 + 37*a^2*b*c*d^7 - 
447*a^3*d^8)/d^8)*(b*x + a) - 15*(7*b^4*c^4*d^4 + 12*a*b^3*c^3*d^5 + 18*a^ 
2*b^2*c^2*d^6 + 28*a^3*b*c*d^7 - 193*a^4*d^8)/d^8)*sqrt(b*x + a) - 15*(7*b 
^6*c^5 + 5*a*b^5*c^4*d + 6*a^2*b^4*c^3*d^2 + 10*a^3*b^3*c^2*d^3 + 35*a^4*b 
^2*c*d^4 - 63*a^5*b*d^5)*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^4))*abs(b)/b^4)/b
 

Mupad [F(-1)]

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

int(x^2*(a + b*x)^(3/2)*(c + d*x)^(1/2),x)
 

Output:

int(x^2*(a + b*x)^(3/2)*(c + d*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.07 \[ \int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx=\frac {45 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{4} b \,d^{5}-30 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} c \,d^{4}-30 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b^{2} d^{5} x -36 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2} d^{3}+18 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} c \,d^{4} x +24 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{3} d^{5} x^{2}+190 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{3} d^{2}-122 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c^{2} d^{3} x +96 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} c \,d^{4} x^{2}+528 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{4} d^{5} x^{3}-105 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{4} d +70 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{3} d^{2} x -56 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c^{2} d^{3} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} c \,d^{4} x^{3}+384 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{5} d^{5} x^{4}-45 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{5} d^{5}+45 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} b c \,d^{4}+30 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b^{2} c^{2} d^{3}+90 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{3} c^{3} d^{2}-225 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{4} c^{4} d +105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{5} c^{5}}{1920 b^{4} d^{5}} \] Input:

int(x^2*(b*x+a)^(3/2)*(d*x+c)^(1/2),x)
 

Output:

(45*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*d**5 - 30*sqrt(c + d*x)*sqrt(a + b* 
x)*a**3*b**2*c*d**4 - 30*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*d**5*x - 36 
*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c**2*d**3 + 18*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**2*b**3*c*d**4*x + 24*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*d**5 
*x**2 + 190*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**3*d**2 - 122*sqrt(c + d* 
x)*sqrt(a + b*x)*a*b**4*c**2*d**3*x + 96*sqrt(c + d*x)*sqrt(a + b*x)*a*b** 
4*c*d**4*x**2 + 528*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*d**5*x**3 - 105*sqr 
t(c + d*x)*sqrt(a + b*x)*b**5*c**4*d + 70*sqrt(c + d*x)*sqrt(a + b*x)*b**5 
*c**3*d**2*x - 56*sqrt(c + d*x)*sqrt(a + b*x)*b**5*c**2*d**3*x**2 + 48*sqr 
t(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 - 45*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 + 45*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 + 
 30*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sq 
rt(a*d - b*c))*a**3*b**2*c**2*d**3 + 90*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt( 
a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**3*c**3*d**2 - 2 
25*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqr 
t(a*d - b*c))*a*b**4*c**4*d + 105*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b* 
x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**5*c**5)/(1920*b**4*d**5)