\(\int x (c+d x)^2 (a x+b x^2)^{3/2} \, dx\) [68]

Optimal result

Integrand size = 22, antiderivative size = 349 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx=\frac {a^4 \left (24 b^2 c^2-28 a b c d+9 a^2 d^2\right ) \sqrt {a x+b x^2}}{1024 b^5}-\frac {a^3 \left (24 b^2 c^2-28 a b c d+9 a^2 d^2\right ) x \sqrt {a x+b x^2}}{1536 b^4}+\frac {a^2 \left (24 b^2 c^2-28 a b c d+9 a^2 d^2\right ) x^2 \sqrt {a x+b x^2}}{1920 b^3}+\frac {11 a \left (24 b^2 c^2-28 a b c d+9 a^2 d^2\right ) x^3 \sqrt {a x+b x^2}}{960 b^2}+\frac {\left (24 b^2 c^2-28 a b c d+9 a^2 d^2\right ) x^4 \sqrt {a x+b x^2}}{120 b}+\frac {d (28 b c-9 a d) x \left (a x+b x^2\right )^{5/2}}{84 b^2}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}-\frac {a^5 \left (24 b^2 c^2-28 a b c d+9 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{1024 b^{11/2}} \] Output:

1/1024*a^4*(9*a^2*d^2-28*a*b*c*d+24*b^2*c^2)*(b*x^2+a*x)^(1/2)/b^5-1/1536* 
a^3*(9*a^2*d^2-28*a*b*c*d+24*b^2*c^2)*x*(b*x^2+a*x)^(1/2)/b^4+1/1920*a^2*( 
9*a^2*d^2-28*a*b*c*d+24*b^2*c^2)*x^2*(b*x^2+a*x)^(1/2)/b^3+11/960*a*(9*a^2 
*d^2-28*a*b*c*d+24*b^2*c^2)*x^3*(b*x^2+a*x)^(1/2)/b^2+1/120*(9*a^2*d^2-28* 
a*b*c*d+24*b^2*c^2)*x^4*(b*x^2+a*x)^(1/2)/b+1/84*d*(-9*a*d+28*b*c)*x*(b*x^ 
2+a*x)^(5/2)/b^2+1/7*d^2*x^2*(b*x^2+a*x)^(5/2)/b-1/1024*a^5*(9*a^2*d^2-28* 
a*b*c*d+24*b^2*c^2)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.89 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (945 a^6 d^2-210 a^5 b d (14 c+3 d x)+192 a^2 b^4 x^2 \left (7 c^2+7 c d x+2 d^2 x^2\right )+56 a^4 b^2 \left (45 c^2+35 c d x+9 d^2 x^2\right )+1024 b^6 x^4 \left (21 c^2+35 c d x+15 d^2 x^2\right )-16 a^3 b^3 x \left (105 c^2+98 c d x+27 d^2 x^2\right )+128 a b^5 x^3 \left (231 c^2+364 c d x+150 d^2 x^2\right )\right )+630 a^5 \left (8 b^2 c^2+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+5880 a^6 b c d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{107520 b^{11/2} \sqrt {x (a+b x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(945*a^6*d^2 - 210*a 
^5*b*d*(14*c + 3*d*x) + 192*a^2*b^4*x^2*(7*c^2 + 7*c*d*x + 2*d^2*x^2) + 56 
*a^4*b^2*(45*c^2 + 35*c*d*x + 9*d^2*x^2) + 1024*b^6*x^4*(21*c^2 + 35*c*d*x 
 + 15*d^2*x^2) - 16*a^3*b^3*x*(105*c^2 + 98*c*d*x + 27*d^2*x^2) + 128*a*b^ 
5*x^3*(231*c^2 + 364*c*d*x + 150*d^2*x^2)) + 630*a^5*(8*b^2*c^2 + 3*a^2*d^ 
2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x])] + 5880*a^6*b*c*d*A 
rcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(107520*b^(11/2)*Sq 
rt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.63, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1262, 27, 1225, 1087, 1087, 1091, 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.

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

\(\Big \downarrow \) 1262

\(\displaystyle \frac {\int \frac {1}{2} x \left (14 b c^2+d (28 b c-9 a d) x\right ) \left (b x^2+a x\right )^{3/2}dx}{7 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x \left (14 b c^2+d (28 b c-9 a d) x\right ) \left (b x^2+a x\right )^{3/2}dx}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {\left (a x+b x^2\right )^{5/2} \left (7 \left (24 b^2 c^2-a d (28 b c-9 a d)\right )+10 b d x (28 b c-9 a d)\right )}{60 b^2}-\frac {7 a \left (24 b^2 c^2-a d (28 b c-9 a d)\right ) \int \left (b x^2+a x\right )^{3/2}dx}{24 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (a x+b x^2\right )^{5/2} \left (7 \left (24 b^2 c^2-a d (28 b c-9 a d)\right )+10 b d x (28 b c-9 a d)\right )}{60 b^2}-\frac {7 a \left (24 b^2 c^2-a d (28 b c-9 a d)\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \int \sqrt {b x^2+a x}dx}{16 b}\right )}{24 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (a x+b x^2\right )^{5/2} \left (7 \left (24 b^2 c^2-a d (28 b c-9 a d)\right )+10 b d x (28 b c-9 a d)\right )}{60 b^2}-\frac {7 a \left (24 b^2 c^2-a d (28 b c-9 a d)\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b}\right )}{24 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {\left (a x+b x^2\right )^{5/2} \left (7 \left (24 b^2 c^2-a d (28 b c-9 a d)\right )+10 b d x (28 b c-9 a d)\right )}{60 b^2}-\frac {7 a \left (24 b^2 c^2-a d (28 b c-9 a d)\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b}\right )}{24 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (a x+b x^2\right )^{5/2} \left (7 \left (24 b^2 c^2-a d (28 b c-9 a d)\right )+10 b d x (28 b c-9 a d)\right )}{60 b^2}-\frac {7 a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right )}{16 b}\right ) \left (24 b^2 c^2-a d (28 b c-9 a d)\right )}{24 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\)

Input:

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

Output:

(d^2*x^2*(a*x + b*x^2)^(5/2))/(7*b) + (((7*(24*b^2*c^2 - a*d*(28*b*c - 9*a 
*d)) + 10*b*d*(28*b*c - 9*a*d)*x)*(a*x + b*x^2)^(5/2))/(60*b^2) - (7*a*(24 
*b^2*c^2 - a*d*(28*b*c - 9*a*d))*(((a + 2*b*x)*(a*x + b*x^2)^(3/2))/(8*b) 
- (3*a^2*(((a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b) - (a^2*ArcTanh[(Sqrt[b]*x) 
/Sqrt[a*x + b*x^2]])/(4*b^(3/2))))/(16*b)))/(24*b^2))/(14*b)
 

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 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])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1262

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.62

method	result	size
pseudoelliptic	\(-\frac {9 \left (a^{5} \left (a^{2} d^{2}-\frac {28}{9} a b c d +\frac {8}{3} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (\frac {1408 \left (\frac {50}{77} d^{2} x^{2}+\frac {52}{33} c d x +c^{2}\right ) x^{3} a \,b^{\frac {11}{2}}}{45}+\frac {1024 x^{4} \left (\frac {5}{7} d^{2} x^{2}+\frac {5}{3} c d x +c^{2}\right ) b^{\frac {13}{2}}}{45}+\left (\frac {8 \left (\frac {1}{5} d^{2} x^{2}+\frac {7}{9} c d x +c^{2}\right ) a^{2} b^{\frac {5}{2}}}{3}-\frac {16 x a \left (\frac {9}{35} d^{2} x^{2}+\frac {14}{15} c d x +c^{2}\right ) b^{\frac {7}{2}}}{9}+\frac {64 x^{2} \left (\frac {2}{7} d^{2} x^{2}+c d x +c^{2}\right ) b^{\frac {9}{2}}}{45}+d \left (\left (-\frac {2 d x}{3}-\frac {28 c}{9}\right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right ) a^{3}\right ) a^{2}\right ) \sqrt {x \left (b x +a \right )}\right )}{1024 b^{\frac {11}{2}}}\)	\(215\)
risch	\(\frac {\left (15360 b^{6} d^{2} x^{6}+19200 a \,b^{5} d^{2} x^{5}+35840 b^{6} c d \,x^{5}+384 a^{2} b^{4} d^{2} x^{4}+46592 a \,b^{5} c d \,x^{4}+21504 b^{6} c^{2} x^{4}-432 a^{3} b^{3} d^{2} x^{3}+1344 a^{2} b^{4} c d \,x^{3}+29568 a \,b^{5} c^{2} x^{3}+504 a^{4} b^{2} d^{2} x^{2}-1568 a^{3} b^{3} c d \,x^{2}+1344 a^{2} b^{4} c^{2} x^{2}-630 a^{5} b \,d^{2} x +1960 a^{4} b^{2} c d x -1680 a^{3} b^{3} c^{2} x +945 a^{6} d^{2}-2940 a^{5} b c d +2520 a^{4} b^{2} c^{2}\right ) x \left (b x +a \right )}{107520 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {a^{5} \left (9 a^{2} d^{2}-28 a b c d +24 b^{2} c^{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2048 b^{\frac {11}{2}}}\)	\(287\)
default	\(c^{2} \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )+d^{2} \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 b}-\frac {9 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}\right )+2 c d \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )\)	\(415\)

Input:

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

Output:

-9/1024/b^(11/2)*(a^5*(a^2*d^2-28/9*a*b*c*d+8/3*b^2*c^2)*arctanh((x*(b*x+a 
))^(1/2)/x/b^(1/2))-(1408/45*(50/77*d^2*x^2+52/33*c*d*x+c^2)*x^3*a*b^(11/2 
)+1024/45*x^4*(5/7*d^2*x^2+5/3*c*d*x+c^2)*b^(13/2)+(8/3*(1/5*d^2*x^2+7/9*c 
*d*x+c^2)*a^2*b^(5/2)-16/9*x*a*(9/35*d^2*x^2+14/15*c*d*x+c^2)*b^(7/2)+64/4 
5*x^2*(2/7*d^2*x^2+c*d*x+c^2)*b^(9/2)+d*((-2/3*d*x-28/9*c)*b^(3/2)+b^(1/2) 
*a*d)*a^3)*a^2)*(x*(b*x+a))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.63 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx=\left [\frac {105 \, {\left (24 \, a^{5} b^{2} c^{2} - 28 \, a^{6} b c d + 9 \, a^{7} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (15360 \, b^{7} d^{2} x^{6} + 2520 \, a^{4} b^{3} c^{2} - 2940 \, a^{5} b^{2} c d + 945 \, a^{6} b d^{2} + 1280 \, {\left (28 \, b^{7} c d + 15 \, a b^{6} d^{2}\right )} x^{5} + 128 \, {\left (168 \, b^{7} c^{2} + 364 \, a b^{6} c d + 3 \, a^{2} b^{5} d^{2}\right )} x^{4} + 48 \, {\left (616 \, a b^{6} c^{2} + 28 \, a^{2} b^{5} c d - 9 \, a^{3} b^{4} d^{2}\right )} x^{3} + 56 \, {\left (24 \, a^{2} b^{5} c^{2} - 28 \, a^{3} b^{4} c d + 9 \, a^{4} b^{3} d^{2}\right )} x^{2} - 70 \, {\left (24 \, a^{3} b^{4} c^{2} - 28 \, a^{4} b^{3} c d + 9 \, a^{5} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{215040 \, b^{6}}, \frac {105 \, {\left (24 \, a^{5} b^{2} c^{2} - 28 \, a^{6} b c d + 9 \, a^{7} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (15360 \, b^{7} d^{2} x^{6} + 2520 \, a^{4} b^{3} c^{2} - 2940 \, a^{5} b^{2} c d + 945 \, a^{6} b d^{2} + 1280 \, {\left (28 \, b^{7} c d + 15 \, a b^{6} d^{2}\right )} x^{5} + 128 \, {\left (168 \, b^{7} c^{2} + 364 \, a b^{6} c d + 3 \, a^{2} b^{5} d^{2}\right )} x^{4} + 48 \, {\left (616 \, a b^{6} c^{2} + 28 \, a^{2} b^{5} c d - 9 \, a^{3} b^{4} d^{2}\right )} x^{3} + 56 \, {\left (24 \, a^{2} b^{5} c^{2} - 28 \, a^{3} b^{4} c d + 9 \, a^{4} b^{3} d^{2}\right )} x^{2} - 70 \, {\left (24 \, a^{3} b^{4} c^{2} - 28 \, a^{4} b^{3} c d + 9 \, a^{5} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{107520 \, b^{6}}\right ] \] Input:

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

Output:

[1/215040*(105*(24*a^5*b^2*c^2 - 28*a^6*b*c*d + 9*a^7*d^2)*sqrt(b)*log(2*b 
*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(15360*b^7*d^2*x^6 + 2520*a^4*b^ 
3*c^2 - 2940*a^5*b^2*c*d + 945*a^6*b*d^2 + 1280*(28*b^7*c*d + 15*a*b^6*d^2 
)*x^5 + 128*(168*b^7*c^2 + 364*a*b^6*c*d + 3*a^2*b^5*d^2)*x^4 + 48*(616*a* 
b^6*c^2 + 28*a^2*b^5*c*d - 9*a^3*b^4*d^2)*x^3 + 56*(24*a^2*b^5*c^2 - 28*a^ 
3*b^4*c*d + 9*a^4*b^3*d^2)*x^2 - 70*(24*a^3*b^4*c^2 - 28*a^4*b^3*c*d + 9*a 
^5*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/b^6, 1/107520*(105*(24*a^5*b^2*c^2 - 28* 
a^6*b*c*d + 9*a^7*d^2)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a 
)) + (15360*b^7*d^2*x^6 + 2520*a^4*b^3*c^2 - 2940*a^5*b^2*c*d + 945*a^6*b* 
d^2 + 1280*(28*b^7*c*d + 15*a*b^6*d^2)*x^5 + 128*(168*b^7*c^2 + 364*a*b^6* 
c*d + 3*a^2*b^5*d^2)*x^4 + 48*(616*a*b^6*c^2 + 28*a^2*b^5*c*d - 9*a^3*b^4* 
d^2)*x^3 + 56*(24*a^2*b^5*c^2 - 28*a^3*b^4*c*d + 9*a^4*b^3*d^2)*x^2 - 70*( 
24*a^3*b^4*c^2 - 28*a^4*b^3*c*d + 9*a^5*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/b^6 
]
 

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.92 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-5*a**3*(a**2*c**2 - 7*a*(2*a**2*c*d + 2*a*b*c**2 - 9*a*(a**2*d 
**2 + 4*a*b*c*d - 11*a*(15*a*b*d**2/14 + 2*b**2*c*d)/(12*b) + b**2*c**2)/( 
10*b))/(8*b))*Piecewise((log(a + 2*sqrt(b)*sqrt(a*x + b*x**2) + 2*b*x)/sqr 
t(b), Ne(a**2/b, 0)), ((a/(2*b) + x)*log(a/(2*b) + x)/sqrt(b*(a/(2*b) + x) 
**2), True))/(16*b**3) + sqrt(a*x + b*x**2)*(5*a**2*(a**2*c**2 - 7*a*(2*a* 
*2*c*d + 2*a*b*c**2 - 9*a*(a**2*d**2 + 4*a*b*c*d - 11*a*(15*a*b*d**2/14 + 
2*b**2*c*d)/(12*b) + b**2*c**2)/(10*b))/(8*b))/(8*b**3) - 5*a*x*(a**2*c**2 
 - 7*a*(2*a**2*c*d + 2*a*b*c**2 - 9*a*(a**2*d**2 + 4*a*b*c*d - 11*a*(15*a* 
b*d**2/14 + 2*b**2*c*d)/(12*b) + b**2*c**2)/(10*b))/(8*b))/(12*b**2) + b*d 
**2*x**6/7 + x**5*(15*a*b*d**2/14 + 2*b**2*c*d)/(6*b) + x**4*(a**2*d**2 + 
4*a*b*c*d - 11*a*(15*a*b*d**2/14 + 2*b**2*c*d)/(12*b) + b**2*c**2)/(5*b) + 
 x**3*(2*a**2*c*d + 2*a*b*c**2 - 9*a*(a**2*d**2 + 4*a*b*c*d - 11*a*(15*a*b 
*d**2/14 + 2*b**2*c*d)/(12*b) + b**2*c**2)/(10*b))/(4*b) + x**2*(a**2*c**2 
 - 7*a*(2*a**2*c*d + 2*a*b*c**2 - 9*a*(a**2*d**2 + 4*a*b*c*d - 11*a*(15*a* 
b*d**2/14 + 2*b**2*c*d)/(12*b) + b**2*c**2)/(10*b))/(8*b))/(3*b)), Ne(b, 0 
)), (2*(c**2*(a*x)**(7/2)/7 + 2*c*d*(a*x)**(9/2)/(9*a) + d**2*(a*x)**(11/2 
)/(11*a**2))/a**2, Ne(a, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.40 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} d^{2} x^{2}}{7 \, b} + \frac {3 \, \sqrt {b x^{2} + a x} a^{3} c^{2} x}{64 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a c^{2} x}{8 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} a^{4} c d x}{128 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} c d x}{48 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} c d x}{3 \, b} + \frac {9 \, \sqrt {b x^{2} + a x} a^{5} d^{2} x}{512 \, b^{4}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} d^{2} x}{64 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a d^{2} x}{28 \, b^{2}} - \frac {3 \, a^{5} c^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} + \frac {7 \, a^{6} c d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{512 \, b^{\frac {9}{2}}} - \frac {9 \, a^{7} d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2048 \, b^{\frac {11}{2}}} + \frac {3 \, \sqrt {b x^{2} + a x} a^{4} c^{2}}{128 \, b^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} c^{2}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} c^{2}}{5 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} a^{5} c d}{256 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} c d}{96 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a c d}{30 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a x} a^{6} d^{2}}{1024 \, b^{5}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4} d^{2}}{128 \, b^{4}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{2} d^{2}}{40 \, b^{3}} \] Input:

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

Output:

1/7*(b*x^2 + a*x)^(5/2)*d^2*x^2/b + 3/64*sqrt(b*x^2 + a*x)*a^3*c^2*x/b^2 - 
 1/8*(b*x^2 + a*x)^(3/2)*a*c^2*x/b - 7/128*sqrt(b*x^2 + a*x)*a^4*c*d*x/b^3 
 + 7/48*(b*x^2 + a*x)^(3/2)*a^2*c*d*x/b^2 + 1/3*(b*x^2 + a*x)^(5/2)*c*d*x/ 
b + 9/512*sqrt(b*x^2 + a*x)*a^5*d^2*x/b^4 - 3/64*(b*x^2 + a*x)^(3/2)*a^3*d 
^2*x/b^3 - 3/28*(b*x^2 + a*x)^(5/2)*a*d^2*x/b^2 - 3/256*a^5*c^2*log(2*b*x 
+ a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 7/512*a^6*c*d*log(2*b*x + a + 
 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) - 9/2048*a^7*d^2*log(2*b*x + a + 2*s 
qrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 3/128*sqrt(b*x^2 + a*x)*a^4*c^2/b^3 - 
 1/16*(b*x^2 + a*x)^(3/2)*a^2*c^2/b^2 + 1/5*(b*x^2 + a*x)^(5/2)*c^2/b - 7/ 
256*sqrt(b*x^2 + a*x)*a^5*c*d/b^4 + 7/96*(b*x^2 + a*x)^(3/2)*a^3*c*d/b^3 - 
 7/30*(b*x^2 + a*x)^(5/2)*a*c*d/b^2 + 9/1024*sqrt(b*x^2 + a*x)*a^6*d^2/b^5 
 - 3/128*(b*x^2 + a*x)^(3/2)*a^4*d^2/b^4 + 3/40*(b*x^2 + a*x)^(5/2)*a^2*d^ 
2/b^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.87 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx=\frac {1}{107520} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, b d^{2} x + \frac {28 \, b^{7} c d + 15 \, a b^{6} d^{2}}{b^{6}}\right )} x + \frac {168 \, b^{7} c^{2} + 364 \, a b^{6} c d + 3 \, a^{2} b^{5} d^{2}}{b^{6}}\right )} x + \frac {3 \, {\left (616 \, a b^{6} c^{2} + 28 \, a^{2} b^{5} c d - 9 \, a^{3} b^{4} d^{2}\right )}}{b^{6}}\right )} x + \frac {7 \, {\left (24 \, a^{2} b^{5} c^{2} - 28 \, a^{3} b^{4} c d + 9 \, a^{4} b^{3} d^{2}\right )}}{b^{6}}\right )} x - \frac {35 \, {\left (24 \, a^{3} b^{4} c^{2} - 28 \, a^{4} b^{3} c d + 9 \, a^{5} b^{2} d^{2}\right )}}{b^{6}}\right )} x + \frac {105 \, {\left (24 \, a^{4} b^{3} c^{2} - 28 \, a^{5} b^{2} c d + 9 \, a^{6} b d^{2}\right )}}{b^{6}}\right )} + \frac {{\left (24 \, a^{5} b^{2} c^{2} - 28 \, a^{6} b c d + 9 \, a^{7} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{2048 \, b^{\frac {11}{2}}} \] Input:

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

Output:

1/107520*sqrt(b*x^2 + a*x)*(2*(4*(2*(8*(10*(12*b*d^2*x + (28*b^7*c*d + 15* 
a*b^6*d^2)/b^6)*x + (168*b^7*c^2 + 364*a*b^6*c*d + 3*a^2*b^5*d^2)/b^6)*x + 
 3*(616*a*b^6*c^2 + 28*a^2*b^5*c*d - 9*a^3*b^4*d^2)/b^6)*x + 7*(24*a^2*b^5 
*c^2 - 28*a^3*b^4*c*d + 9*a^4*b^3*d^2)/b^6)*x - 35*(24*a^3*b^4*c^2 - 28*a^ 
4*b^3*c*d + 9*a^5*b^2*d^2)/b^6)*x + 105*(24*a^4*b^3*c^2 - 28*a^5*b^2*c*d + 
 9*a^6*b*d^2)/b^6) + 1/2048*(24*a^5*b^2*c^2 - 28*a^6*b*c*d + 9*a^7*d^2)*lo 
g(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.29 \[ \int x (c+d x)^2 \left (a x+b x^2\right )^{3/2} \, dx=\frac {945 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,d^{2}-2940 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c d -630 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} d^{2} x +2520 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c^{2}+1960 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c d x +504 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} d^{2} x^{2}-1680 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{2} x -1568 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c d \,x^{2}-432 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} d^{2} x^{3}+1344 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{2} x^{2}+1344 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c d \,x^{3}+384 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} d^{2} x^{4}+29568 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{2} x^{3}+46592 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c d \,x^{4}+19200 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} d^{2} x^{5}+21504 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c^{2} x^{4}+35840 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c d \,x^{5}+15360 \sqrt {x}\, \sqrt {b x +a}\, b^{7} d^{2} x^{6}-945 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{7} d^{2}+2940 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6} b c d -2520 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} b^{2} c^{2}}{107520 b^{6}} \] Input:

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

Output:

(945*sqrt(x)*sqrt(a + b*x)*a**6*b*d**2 - 2940*sqrt(x)*sqrt(a + b*x)*a**5*b 
**2*c*d - 630*sqrt(x)*sqrt(a + b*x)*a**5*b**2*d**2*x + 2520*sqrt(x)*sqrt(a 
 + b*x)*a**4*b**3*c**2 + 1960*sqrt(x)*sqrt(a + b*x)*a**4*b**3*c*d*x + 504* 
sqrt(x)*sqrt(a + b*x)*a**4*b**3*d**2*x**2 - 1680*sqrt(x)*sqrt(a + b*x)*a** 
3*b**4*c**2*x - 1568*sqrt(x)*sqrt(a + b*x)*a**3*b**4*c*d*x**2 - 432*sqrt(x 
)*sqrt(a + b*x)*a**3*b**4*d**2*x**3 + 1344*sqrt(x)*sqrt(a + b*x)*a**2*b**5 
*c**2*x**2 + 1344*sqrt(x)*sqrt(a + b*x)*a**2*b**5*c*d*x**3 + 384*sqrt(x)*s 
qrt(a + b*x)*a**2*b**5*d**2*x**4 + 29568*sqrt(x)*sqrt(a + b*x)*a*b**6*c**2 
*x**3 + 46592*sqrt(x)*sqrt(a + b*x)*a*b**6*c*d*x**4 + 19200*sqrt(x)*sqrt(a 
 + b*x)*a*b**6*d**2*x**5 + 21504*sqrt(x)*sqrt(a + b*x)*b**7*c**2*x**4 + 35 
840*sqrt(x)*sqrt(a + b*x)*b**7*c*d*x**5 + 15360*sqrt(x)*sqrt(a + b*x)*b**7 
*d**2*x**6 - 945*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a* 
*7*d**2 + 2940*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**6 
*b*c*d - 2520*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**5* 
b**2*c**2)/(107520*b**6)