\(\int (c+d x) (a+b x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [73]

Optimal result

Integrand size = 32, antiderivative size = 272 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a \left (48 A b^2 c-a (8 b (c C+B d)-3 a d D)\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (48 A b^2 c-a (8 b (c C+B d)-3 a d D)\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {(b B c+A b d-a C d-a c D) \left (a+b x^2\right )^{5/2}}{5 b^2}+\frac {(8 b (c C+B d)-3 a d D) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d D x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {(C d+c D) \left (a+b x^2\right )^{7/2}}{7 b^2}+\frac {a^2 \left (48 A b^2 c-a (8 b (c C+B d)-3 a d D)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:

1/128*a*(48*A*b^2*c-a*(8*b*(B*d+C*c)-3*D*a*d))*x*(b*x^2+a)^(1/2)/b^2+1/192 
*(48*A*b^2*c-a*(8*b*(B*d+C*c)-3*D*a*d))*x*(b*x^2+a)^(3/2)/b^2+1/5*(A*b*d+B 
*b*c-C*a*d-D*a*c)*(b*x^2+a)^(5/2)/b^2+1/48*(8*b*(B*d+C*c)-3*D*a*d)*x*(b*x^ 
2+a)^(5/2)/b^2+1/8*d*D*x^3*(b*x^2+a)^(5/2)/b+1/7*(C*d+D*c)*(b*x^2+a)^(7/2) 
/b^2+1/128*a^2*(48*A*b^2*c-a*(8*b*(B*d+C*c)-3*D*a*d))*arctanh(b^(1/2)*x/(b 
*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.97 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (-3 a^3 (256 C d+256 c D+105 d D x)+6 a^2 b \left (448 A d+28 B (16 c+5 d x)+x \left (140 c C+64 C d x+64 c D x+35 d D x^2\right )\right )+8 a b^2 x \left (42 A (25 c+16 d x)+x \left (14 B (48 c+35 d x)+x \left (490 c C+384 C d x+384 c D x+315 d D x^2\right )\right )\right )+16 b^3 x^3 (42 A (5 c+4 d x)+x (28 B (6 c+5 d x)+5 x (4 c (7 C+6 D x)+3 d x (8 C+7 D x))))\right )-105 a^2 \left (48 A b^2 c+a (-8 b (c C+B d)+3 a d D)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{13440 b^{5/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(-3*a^3*(256*C*d + 256*c*D + 105*d*D*x) + 6*a^2*b 
*(448*A*d + 28*B*(16*c + 5*d*x) + x*(140*c*C + 64*C*d*x + 64*c*D*x + 35*d* 
D*x^2)) + 8*a*b^2*x*(42*A*(25*c + 16*d*x) + x*(14*B*(48*c + 35*d*x) + x*(4 
90*c*C + 384*C*d*x + 384*c*D*x + 315*d*D*x^2))) + 16*b^3*x^3*(42*A*(5*c + 
4*d*x) + x*(28*B*(6*c + 5*d*x) + 5*x*(4*c*(7*C + 6*D*x) + 3*d*x*(8*C + 7*D 
*x))))) - 105*a^2*(48*A*b^2*c + a*(-8*b*(c*C + B*d) + 3*a*d*D))*Log[-(Sqrt 
[b]*x) + Sqrt[a + b*x^2]])/(13440*b^(5/2))
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2185, 2185, 27, 676, 211, 211, 224, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int (c+d x) \left (b x^2+a\right )^{3/2} \left (b (8 C d-13 c D) x^2 d^2+(8 A b d-3 a c D) d^2+\left (-5 b D c^2+8 b B d^2-3 a d^2 D\right ) x d\right )dx}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int b d^3 (c+d x) \left (d (56 A b d-16 a C d+5 a c D)-\left (21 a D d^2+b \left (-30 D c^2+40 C d c-56 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b d^2}+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} d \int (c+d x) \left (d (56 A b d-16 a C d+5 a c D)-\left (21 a D d^2+b \left (-30 D c^2+40 C d c-56 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{7} d \left (\frac {7 d^2 \left (48 A b^2 c-a (8 b (B d+c C)-3 a d D)\right ) \int \left (b x^2+a\right )^{3/2}dx}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (c D+C d)+b \left (-28 A d^3-28 B c d^2-15 c^3 D+20 c^2 C d\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a d^2 D+b \left (-56 B d^2-30 c^2 D+40 c C d\right )\right )}{6 b}\right )+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{7} d \left (\frac {7 d^2 \left (48 A b^2 c-a (8 b (B d+c C)-3 a d D)\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (c D+C d)+b \left (-28 A d^3-28 B c d^2-15 c^3 D+20 c^2 C d\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a d^2 D+b \left (-56 B d^2-30 c^2 D+40 c C d\right )\right )}{6 b}\right )+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{7} d \left (\frac {7 d^2 \left (48 A b^2 c-a (8 b (B d+c C)-3 a d D)\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (c D+C d)+b \left (-28 A d^3-28 B c d^2-15 c^3 D+20 c^2 C d\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a d^2 D+b \left (-56 B d^2-30 c^2 D+40 c C d\right )\right )}{6 b}\right )+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{7} d \left (\frac {7 d^2 \left (48 A b^2 c-a (8 b (B d+c C)-3 a d D)\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (c D+C d)+b \left (-28 A d^3-28 B c d^2-15 c^3 D+20 c^2 C d\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a d^2 D+b \left (-56 B d^2-30 c^2 D+40 c C d\right )\right )}{6 b}\right )+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{7} d \left (\frac {7 d^2 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (48 A b^2 c-a (8 b (B d+c C)-3 a d D)\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (c D+C d)+b \left (-28 A d^3-28 B c d^2-15 c^3 D+20 c^2 C d\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a d^2 D+b \left (-56 B d^2-30 c^2 D+40 c C d\right )\right )}{6 b}\right )+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 C d-13 c D)}{8 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}\)

Input:

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

Output:

(D*(c + d*x)^3*(a + b*x^2)^(5/2))/(8*b*d^2) + ((d*(8*C*d - 13*c*D)*(c + d* 
x)^2*(a + b*x^2)^(5/2))/7 + (d*((-2*(8*a*d^2*(C*d + c*D) + b*(20*c^2*C*d - 
 28*B*c*d^2 - 28*A*d^3 - 15*c^3*D))*(a + b*x^2)^(5/2))/(5*b) - (d*(21*a*d^ 
2*D + b*(40*c*C*d - 56*B*d^2 - 30*c^2*D))*x*(a + b*x^2)^(5/2))/(6*b) + (7* 
d^2*(48*A*b^2*c - a*(8*b*(c*C + B*d) - 3*a*d*D))*((x*(a + b*x^2)^(3/2))/4 
+ (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/( 
2*Sqrt[b])))/4))/(6*b)))/7)/(8*b*d^3)
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 676

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

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.10

method	result	size
default	\(A c \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\frac {\left (A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+\left (B d +C c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+\left (C d +D c \right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+D d \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )\)	\(299\)

Input:

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

Output:

A*c*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b 
^(1/2)*x+(b*x^2+a)^(1/2))))+1/5*(A*d+B*c)/b*(b*x^2+a)^(5/2)+(B*d+C*c)*(1/6 
*x*(b*x^2+a)^(5/2)/b-1/6*a/b*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a) 
^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))))+(C*d+D*c)*(1/7*x^2*( 
b*x^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+D*d*(1/8*x^3*(b*x^2+a)^(5/2)/ 
b-3/8*a/b*(1/6*x*(b*x^2+a)^(5/2)/b-1/6*a/b*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1 
/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 689, normalized size of antiderivative = 2.53 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

[1/26880*(105*(8*(C*a^3*b - 6*A*a^2*b^2)*c - (3*D*a^4 - 8*B*a^3*b)*d)*sqrt 
(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(1680*D*b^4*d*x^7 
+ 1920*(D*b^4*c + C*b^4*d)*x^6 + 280*(8*C*b^4*c + (9*D*a*b^3 + 8*B*b^4)*d) 
*x^5 + 384*((8*D*a*b^3 + 7*B*b^4)*c + (8*C*a*b^3 + 7*A*b^4)*d)*x^4 + 70*(8 
*(7*C*a*b^3 + 6*A*b^4)*c + (3*D*a^2*b^2 + 56*B*a*b^3)*d)*x^3 + 384*((D*a^2 
*b^2 + 14*B*a*b^3)*c + (C*a^2*b^2 + 14*A*a*b^3)*d)*x^2 - 384*(2*D*a^3*b - 
7*B*a^2*b^2)*c - 384*(2*C*a^3*b - 7*A*a^2*b^2)*d + 105*(8*(C*a^2*b^2 + 10* 
A*a*b^3)*c - (3*D*a^3*b - 8*B*a^2*b^2)*d)*x)*sqrt(b*x^2 + a))/b^3, 1/13440 
*(105*(8*(C*a^3*b - 6*A*a^2*b^2)*c - (3*D*a^4 - 8*B*a^3*b)*d)*sqrt(-b)*arc 
tan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (1680*D*b^4*d*x^7 + 1920*(D*b^4*c + C*b^ 
4*d)*x^6 + 280*(8*C*b^4*c + (9*D*a*b^3 + 8*B*b^4)*d)*x^5 + 384*((8*D*a*b^3 
 + 7*B*b^4)*c + (8*C*a*b^3 + 7*A*b^4)*d)*x^4 + 70*(8*(7*C*a*b^3 + 6*A*b^4) 
*c + (3*D*a^2*b^2 + 56*B*a*b^3)*d)*x^3 + 384*((D*a^2*b^2 + 14*B*a*b^3)*c + 
 (C*a^2*b^2 + 14*A*a*b^3)*d)*x^2 - 384*(2*D*a^3*b - 7*B*a^2*b^2)*c - 384*( 
2*C*a^3*b - 7*A*a^2*b^2)*d + 105*(8*(C*a^2*b^2 + 10*A*a*b^3)*c - (3*D*a^3* 
b - 8*B*a^2*b^2)*d)*x)*sqrt(b*x^2 + a))/b^3]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (262) = 524\).

Time = 0.82 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.56 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {D b d x^{7}}{8} + \frac {x^{6} \left (C b^{2} d + D b^{2} c\right )}{7 b} + \frac {x^{5} \left (B b^{2} d + C b^{2} c + \frac {9 D a b d}{8}\right )}{6 b} + \frac {x^{4} \left (A b^{2} d + B b^{2} c + 2 C a b d + 2 D a b c - \frac {6 a \left (C b^{2} d + D b^{2} c\right )}{7 b}\right )}{5 b} + \frac {x^{3} \left (A b^{2} c + 2 B a b d + 2 C a b c + D a^{2} d - \frac {5 a \left (B b^{2} d + C b^{2} c + \frac {9 D a b d}{8}\right )}{6 b}\right )}{4 b} + \frac {x^{2} \cdot \left (2 A a b d + 2 B a b c + C a^{2} d + D a^{2} c - \frac {4 a \left (A b^{2} d + B b^{2} c + 2 C a b d + 2 D a b c - \frac {6 a \left (C b^{2} d + D b^{2} c\right )}{7 b}\right )}{5 b}\right )}{3 b} + \frac {x \left (2 A a b c + B a^{2} d + C a^{2} c - \frac {3 a \left (A b^{2} c + 2 B a b d + 2 C a b c + D a^{2} d - \frac {5 a \left (B b^{2} d + C b^{2} c + \frac {9 D a b d}{8}\right )}{6 b}\right )}{4 b}\right )}{2 b} + \frac {A a^{2} d + B a^{2} c - \frac {2 a \left (2 A a b d + 2 B a b c + C a^{2} d + D a^{2} c - \frac {4 a \left (A b^{2} d + B b^{2} c + 2 C a b d + 2 D a b c - \frac {6 a \left (C b^{2} d + D b^{2} c\right )}{7 b}\right )}{5 b}\right )}{3 b}}{b}\right ) + \left (A a^{2} c - \frac {a \left (2 A a b c + B a^{2} d + C a^{2} c - \frac {3 a \left (A b^{2} c + 2 B a b d + 2 C a b c + D a^{2} d - \frac {5 a \left (B b^{2} d + C b^{2} c + \frac {9 D a b d}{8}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (A c x + \frac {D d x^{5}}{5} + \frac {x^{4} \left (C d + D c\right )}{4} + \frac {x^{3} \left (B d + C c\right )}{3} + \frac {x^{2} \left (A d + B c\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((d*x+c)*(b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Piecewise((sqrt(a + b*x**2)*(D*b*d*x**7/8 + x**6*(C*b**2*d + D*b**2*c)/(7* 
b) + x**5*(B*b**2*d + C*b**2*c + 9*D*a*b*d/8)/(6*b) + x**4*(A*b**2*d + B*b 
**2*c + 2*C*a*b*d + 2*D*a*b*c - 6*a*(C*b**2*d + D*b**2*c)/(7*b))/(5*b) + x 
**3*(A*b**2*c + 2*B*a*b*d + 2*C*a*b*c + D*a**2*d - 5*a*(B*b**2*d + C*b**2* 
c + 9*D*a*b*d/8)/(6*b))/(4*b) + x**2*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d + D 
*a**2*c - 4*a*(A*b**2*d + B*b**2*c + 2*C*a*b*d + 2*D*a*b*c - 6*a*(C*b**2*d 
 + D*b**2*c)/(7*b))/(5*b))/(3*b) + x*(2*A*a*b*c + B*a**2*d + C*a**2*c - 3* 
a*(A*b**2*c + 2*B*a*b*d + 2*C*a*b*c + D*a**2*d - 5*a*(B*b**2*d + C*b**2*c 
+ 9*D*a*b*d/8)/(6*b))/(4*b))/(2*b) + (A*a**2*d + B*a**2*c - 2*a*(2*A*a*b*d 
 + 2*B*a*b*c + C*a**2*d + D*a**2*c - 4*a*(A*b**2*d + B*b**2*c + 2*C*a*b*d 
+ 2*D*a*b*c - 6*a*(C*b**2*d + D*b**2*c)/(7*b))/(5*b))/(3*b))/b) + (A*a**2* 
c - a*(2*A*a*b*c + B*a**2*d + C*a**2*c - 3*a*(A*b**2*c + 2*B*a*b*d + 2*C*a 
*b*c + D*a**2*d - 5*a*(B*b**2*d + C*b**2*c + 9*D*a*b*d/8)/(6*b))/(4*b))/(2 
*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)) 
, (x*log(x)/sqrt(b*x**2), True)), Ne(b, 0)), (a**(3/2)*(A*c*x + D*d*x**5/5 
 + x**4*(C*d + D*c)/4 + x**3*(B*d + C*c)/3 + x**2*(A*d + B*c)/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.17 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D d x^{3}}{8 \, b} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A c x + \frac {3}{8} \, \sqrt {b x^{2} + a} A a c x - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D a d x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D a^{2} d x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} D a^{3} d x}{128 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (D c + C d\right )} x^{2}}{7 \, b} + \frac {3 \, A a^{2} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {3 \, D a^{4} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B c}{5 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A d}{5 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )} a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} {\left (C c + B d\right )} a^{2} x}{16 \, b} - \frac {{\left (C c + B d\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (D c + C d\right )} a}{35 \, b^{2}} \] Input:

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

Output:

1/8*(b*x^2 + a)^(5/2)*D*d*x^3/b + 1/4*(b*x^2 + a)^(3/2)*A*c*x + 3/8*sqrt(b 
*x^2 + a)*A*a*c*x - 1/16*(b*x^2 + a)^(5/2)*D*a*d*x/b^2 + 1/64*(b*x^2 + a)^ 
(3/2)*D*a^2*d*x/b^2 + 3/128*sqrt(b*x^2 + a)*D*a^3*d*x/b^2 + 1/7*(b*x^2 + a 
)^(5/2)*(D*c + C*d)*x^2/b + 3/8*A*a^2*c*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 3 
/128*D*a^4*d*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 1/5*(b*x^2 + a)^(5/2)*B*c/b 
+ 1/5*(b*x^2 + a)^(5/2)*A*d/b + 1/6*(b*x^2 + a)^(5/2)*(C*c + B*d)*x/b - 1/ 
24*(b*x^2 + a)^(3/2)*(C*c + B*d)*a*x/b - 1/16*sqrt(b*x^2 + a)*(C*c + B*d)* 
a^2*x/b - 1/16*(C*c + B*d)*a^3*arcsinh(b*x/sqrt(a*b))/b^(3/2) - 2/35*(b*x^ 
2 + a)^(5/2)*(D*c + C*d)*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.28 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{13440} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, D b d x + \frac {8 \, {\left (D b^{7} c + C b^{7} d\right )}}{b^{6}}\right )} x + \frac {7 \, {\left (8 \, C b^{7} c + 9 \, D a b^{6} d + 8 \, B b^{7} d\right )}}{b^{6}}\right )} x + \frac {48 \, {\left (8 \, D a b^{6} c + 7 \, B b^{7} c + 8 \, C a b^{6} d + 7 \, A b^{7} d\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (56 \, C a b^{6} c + 48 \, A b^{7} c + 3 \, D a^{2} b^{5} d + 56 \, B a b^{6} d\right )}}{b^{6}}\right )} x + \frac {192 \, {\left (D a^{2} b^{5} c + 14 \, B a b^{6} c + C a^{2} b^{5} d + 14 \, A a b^{6} d\right )}}{b^{6}}\right )} x + \frac {105 \, {\left (8 \, C a^{2} b^{5} c + 80 \, A a b^{6} c - 3 \, D a^{3} b^{4} d + 8 \, B a^{2} b^{5} d\right )}}{b^{6}}\right )} x - \frac {384 \, {\left (2 \, D a^{3} b^{4} c - 7 \, B a^{2} b^{5} c + 2 \, C a^{3} b^{4} d - 7 \, A a^{2} b^{5} d\right )}}{b^{6}}\right )} + \frac {{\left (8 \, C a^{3} b c - 48 \, A a^{2} b^{2} c - 3 \, D a^{4} d + 8 \, B a^{3} b d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:

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

Output:

1/13440*sqrt(b*x^2 + a)*((2*((4*(5*(6*(7*D*b*d*x + 8*(D*b^7*c + C*b^7*d)/b 
^6)*x + 7*(8*C*b^7*c + 9*D*a*b^6*d + 8*B*b^7*d)/b^6)*x + 48*(8*D*a*b^6*c + 
 7*B*b^7*c + 8*C*a*b^6*d + 7*A*b^7*d)/b^6)*x + 35*(56*C*a*b^6*c + 48*A*b^7 
*c + 3*D*a^2*b^5*d + 56*B*a*b^6*d)/b^6)*x + 192*(D*a^2*b^5*c + 14*B*a*b^6* 
c + C*a^2*b^5*d + 14*A*a*b^6*d)/b^6)*x + 105*(8*C*a^2*b^5*c + 80*A*a*b^6*c 
 - 3*D*a^3*b^4*d + 8*B*a^2*b^5*d)/b^6)*x - 384*(2*D*a^3*b^4*c - 7*B*a^2*b^ 
5*c + 2*C*a^3*b^4*d - 7*A*a^2*b^5*d)/b^6) + 1/128*(8*C*a^3*b*c - 48*A*a^2* 
b^2*c - 3*D*a^4*d + 8*B*a^3*b*d)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^ 
(5/2)
 

Mupad [F(-1)]

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

int((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

int((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [B] (verification not implemented)

Time = 2.59 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.96 \[ \int (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {2688 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d +2688 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c +2688 \sqrt {b \,x^{2}+a}\, b^{5} c \,x^{4}+2240 \sqrt {b \,x^{2}+a}\, b^{5} d \,x^{5}+2240 \sqrt {b \,x^{2}+a}\, b^{4} c^{2} x^{5}+1680 \sqrt {b \,x^{2}+a}\, b^{4} d^{2} x^{7}+315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d^{2}-1536 \sqrt {b \,x^{2}+a}\, a^{3} b c d -315 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{2} x +8400 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c x +5376 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{2}+840 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d x +840 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} x +210 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{2} x^{3}+3360 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{3}+5376 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{2}+2688 \sqrt {b \,x^{2}+a}\, a \,b^{4} d \,x^{4}+3920 \sqrt {b \,x^{2}+a}\, a \,b^{4} d \,x^{3}+3920 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} x^{3}+2520 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{2} x^{5}+3840 \sqrt {b \,x^{2}+a}\, b^{4} c d \,x^{6}+5040 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} c -840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} d -840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b \,c^{2}+768 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c d \,x^{2}+6144 \sqrt {b \,x^{2}+a}\, a \,b^{3} c d \,x^{4}}{13440 b^{3}} \] Input:

int((d*x+c)*(b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(2688*sqrt(a + b*x**2)*a**3*b**2*d - 1536*sqrt(a + b*x**2)*a**3*b*c*d - 31 
5*sqrt(a + b*x**2)*a**3*b*d**2*x + 8400*sqrt(a + b*x**2)*a**2*b**3*c*x + 2 
688*sqrt(a + b*x**2)*a**2*b**3*c + 5376*sqrt(a + b*x**2)*a**2*b**3*d*x**2 
+ 840*sqrt(a + b*x**2)*a**2*b**3*d*x + 840*sqrt(a + b*x**2)*a**2*b**2*c**2 
*x + 768*sqrt(a + b*x**2)*a**2*b**2*c*d*x**2 + 210*sqrt(a + b*x**2)*a**2*b 
**2*d**2*x**3 + 3360*sqrt(a + b*x**2)*a*b**4*c*x**3 + 5376*sqrt(a + b*x**2 
)*a*b**4*c*x**2 + 2688*sqrt(a + b*x**2)*a*b**4*d*x**4 + 3920*sqrt(a + b*x* 
*2)*a*b**4*d*x**3 + 3920*sqrt(a + b*x**2)*a*b**3*c**2*x**3 + 6144*sqrt(a + 
 b*x**2)*a*b**3*c*d*x**4 + 2520*sqrt(a + b*x**2)*a*b**3*d**2*x**5 + 2688*s 
qrt(a + b*x**2)*b**5*c*x**4 + 2240*sqrt(a + b*x**2)*b**5*d*x**5 + 2240*sqr 
t(a + b*x**2)*b**4*c**2*x**5 + 3840*sqrt(a + b*x**2)*b**4*c*d*x**6 + 1680* 
sqrt(a + b*x**2)*b**4*d**2*x**7 + 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt 
(b)*x)/sqrt(a))*a**4*d**2 + 5040*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x 
)/sqrt(a))*a**3*b**2*c - 840*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sq 
rt(a))*a**3*b**2*d - 840*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a 
))*a**3*b*c**2)/(13440*b**3)