3.198 \(\int (g+h x) (a+b x+c x^2)^{3/2} (d+e x+f x^2) \, dx\)
Optimal. Leaf size=418 \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{840 c^3 h}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{384 c^4}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{1024 c^5}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{2048 c^{11/2}}+\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h} \]
[Out]
-((b^2 - 4*a*c)*(48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*
g + e*h)))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) + ((48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g +
3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*g + e*h)))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^4) + (f*(g
+ h*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c*h) + ((63*b^2*f*h^2 - 24*c^2*(5*f*g^2 - 7*h*(e*g + d*h)) - 2*c*h*(24*a*
f*h + 49*b*(f*g + e*h)) - 10*c*h*(10*c*f*g - 14*c*e*h + 9*b*f*h)*x)*(a + b*x + c*x^2)^(5/2))/(840*c^3*h) + ((b
^2 - 4*a*c)^2*(48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*g
+ e*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))
________________________________________________________________________________________
Rubi [A] time = 0.64543, antiderivative size = 418, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{1653, 779, 612, 621, 206} \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left (5 f g^2-7 h (d h+e g)\right )\right )}{840 c^3 h}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{384 c^4}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{1024 c^5}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right )}{2048 c^{11/2}}+\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h} \]
Antiderivative was successfully verified.
[In]
Int[(g + h*x)*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]
[Out]
-((b^2 - 4*a*c)*(48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*
g + e*h)))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) + ((48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g +
3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*g + e*h)))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^4) + (f*(g
+ h*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c*h) + ((63*b^2*f*h^2 - 24*c^2*(5*f*g^2 - 7*h*(e*g + d*h)) - 2*c*h*(24*a*
f*h + 49*b*(f*g + e*h)) - 10*c*h*(10*c*f*g - 14*c*e*h + 9*b*f*h)*x)*(a + b*x + c*x^2)^(5/2))/(840*c^3*h) + ((b
^2 - 4*a*c)^2*(48*c^3*d*g - 9*b^3*f*h - 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) + 2*b*c*(6*a*f*h + 7*b*(f*g
+ e*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))
Rule 1653
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(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 + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Rule 779
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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Rule 612
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] - Dist[(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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
Rule 621
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int (g+h x) \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac{\int (g+h x) \left (-\frac{1}{2} h (5 b f g-14 c d h+4 a f h)-\frac{1}{2} h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c h^2}\\ &=\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac{\left (63 b^2 f h^2-24 c^2 \left (5 f g^2-7 h (e g+d h)\right )-2 c h (24 a f h+49 b (f g+e h))-10 c h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3 h}+\frac{\left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=\frac{\left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac{\left (63 b^2 f h^2-24 c^2 \left (5 f g^2-7 h (e g+d h)\right )-2 c h (24 a f h+49 b (f g+e h))-10 c h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3 h}-\frac{\left (\left (b^2-4 a c\right ) \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{256 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^5}+\frac{\left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac{\left (63 b^2 f h^2-24 c^2 \left (5 f g^2-7 h (e g+d h)\right )-2 c h (24 a f h+49 b (f g+e h))-10 c h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3 h}+\frac{\left (\left (b^2-4 a c\right )^2 \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac{\left (b^2-4 a c\right ) \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^5}+\frac{\left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac{\left (63 b^2 f h^2-24 c^2 \left (5 f g^2-7 h (e g+d h)\right )-2 c h (24 a f h+49 b (f g+e h))-10 c h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3 h}+\frac{\left (\left (b^2-4 a c\right )^2 \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac{\left (b^2-4 a c\right ) \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^5}+\frac{\left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{f (g+h x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c h}+\frac{\left (63 b^2 f h^2-24 c^2 \left (5 f g^2-7 h (e g+d h)\right )-2 c h (24 a f h+49 b (f g+e h))-10 c h (10 c f g-14 c e h+9 b f h) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3 h}+\frac{\left (b^2-4 a c\right )^2 \left (48 c^3 d g-9 b^3 f h-8 c^2 (3 b e g+a f g+3 b d h+a e h)+2 b c (6 a f h+7 b (f g+e h))\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.806003, size = 285, normalized size = 0.68 \[ \frac{\frac{(a+x (b+c x))^{5/2} \left (-2 c h (24 a f h+b (49 e h+49 f g+45 f h x))+63 b^2 f h^2-4 c^2 (5 f g (6 g+5 h x)-7 h (6 d h+6 e g+5 e h x))\right )}{120 c^2}-\frac{7 h \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right ) \left (8 c^2 (a e h+a f g+3 b d h+3 b e g)-2 b c (6 a f h+7 b (e h+f g))+9 b^3 f h-48 c^3 d g\right )}{6144 c^{9/2}}+f (g+h x)^2 (a+x (b+c x))^{5/2}}{7 c h} \]
Antiderivative was successfully verified.
[In]
Integrate[(g + h*x)*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]
[Out]
(f*(g + h*x)^2*(a + x*(b + c*x))^(5/2) + ((a + x*(b + c*x))^(5/2)*(63*b^2*f*h^2 - 4*c^2*(5*f*g*(6*g + 5*h*x) -
7*h*(6*e*g + 6*d*h + 5*e*h*x)) - 2*c*h*(24*a*f*h + b*(49*f*g + 49*e*h + 45*f*h*x))))/(120*c^2) - (7*h*(-48*c^
3*d*g + 9*b^3*f*h + 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) - 2*b*c*(6*a*f*h + 7*b*(f*g + e*h)))*(2*Sqrt[c]*
(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*
c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(6144*c^(9/2)))/(7*c*h)
________________________________________________________________________________________
Maple [B] time = 0.059, size = 2026, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((h*x+g)*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x)
[Out]
1/8*d*g/c*(c*x^2+b*x+a)^(3/2)*b+3/8*d*g*(c*x^2+b*x+a)^(1/2)*x*a+1/16*h*f*b/c^2*a*x*(c*x^2+b*x+a)^(3/2)-3/16*b/
c*(c*x^2+b*x+a)^(1/2)*x*a*d*h-3/16*b/c*(c*x^2+b*x+a)^(1/2)*x*a*e*g+3/32*h*f*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x-3/
32*h*f*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a+1/4*d*g*x*(c*x^2+b*x+a)^(3/2)+1/5*(c*x^2+b*x+a)^(5/2)/c*d*h+1/5*(c*x^2+
b*x+a)^(5/2)/c*e*g-3/64*d*g/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d*g/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))*a^2+3/128*d*g/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4+3/128*b^4/c^3*(c*x^2+b*x+a)^(1/2)*
e*g+1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*e*h+1/8*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a*f*g+7/1024*b^6/c^(9/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*g-1/16*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h-7/51
2*b^5/c^4*(c*x^2+b*x+a)^(1/2)*f*g+7/192*b^3/c^3*(c*x^2+b*x+a)^(3/2)*f*g-1/16*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*f*g-7/512*b^5/c^4*(c*x^2+b*x+a)^(1/2)*e*h+1/6*x*(c*x^2+b*x+a)^(5/2)/c*f*g-1/16*b^2/c^2*
(c*x^2+b*x+a)^(3/2)*e*g+3/128*b^4/c^3*(c*x^2+b*x+a)^(1/2)*d*h-3/256*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*d*h-3/256*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*g+1/7*h*f*x^2*(c*x^2+b*x+a)^
(5/2)/c-3/128*h*f*b^4/c^4*(c*x^2+b*x+a)^(3/2)+9/1024*h*f*b^6/c^5*(c*x^2+b*x+a)^(1/2)+3/40*h*f*b^2/c^3*(c*x^2+b
*x+a)^(5/2)-2/35*h*f*a/c^2*(c*x^2+b*x+a)^(5/2)-9/2048*h*f*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))-1/16*b^2/c^2*(c*x^2+b*x+a)^(3/2)*d*h+7/192*b^3/c^3*(c*x^2+b*x+a)^(3/2)*e*h+1/6*x*(c*x^2+b*x+a)^(5/2)/c*e*
h+7/1024*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e*h-7/60*b/c^2*(c*x^2+b*x+a)^(5/2)*e*h-7/60*b
/c^2*(c*x^2+b*x+a)^(5/2)*f*g+3/64*h*f*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)-3/28*h*f*b/c^2*x*(c*x^2+b*x+a)^(5/2)-3/6
4*h*f*b^3/c^3*x*(c*x^2+b*x+a)^(3/2)+9/512*h*f*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x-3/64*h*f*b^4/c^4*(c*x^2+b*x+a)^(1/
2)*a-15/256*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f*g-1/24*a/c*x*(c*x^2+b*x+a)^(3/2)*e*h+9
/64*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*f*g+1/16*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a*f*g+9/6
4*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*e*h-1/32*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b*e*h-3/16*
d*g/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a-3/32*d*g/c*(c*x^2+b*x+a)^(1/2)*x*b^2+3/16*d*g/c*
(c*x^2+b*x+a)^(1/2)*b*a-3/32*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a*d*h-3/32*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a*e*g-3/16*b/c
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*h-3/16*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a^2*e*g+3/32*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*h+3/32*b^3/c^(5/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*g-1/8*b/c*x*(c*x^2+b*x+a)^(3/2)*d*h-1/8*b/c*x*(c*x^2+b*x+a)^(3/2)*e*g+3/
64*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*d*h+3/64*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*e*g-1/32*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b
*f*g-1/48*a/c^2*(c*x^2+b*x+a)^(3/2)*b*e*h-1/16*a^2/c*(c*x^2+b*x+a)^(1/2)*x*e*h-15/256*b^4/c^(7/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e*h+1/32*h*f*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)+3/32*h*f*b/c^(5/2)*a^3*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/48*a/c^2*(c*x^2+b*x+a)^(3/2)*b*f*g+21/512*h*f*b^5/c^(9/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*a-15/128*h*f*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+7/96*b^2/
c^2*x*(c*x^2+b*x+a)^(3/2)*e*h+7/96*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)*f*g-1/24*a/c*x*(c*x^2+b*x+a)^(3/2)*f*g-7/256*
b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*f*g-1/16*a^2/c*(c*x^2+b*x+a)^(1/2)*x*f*g+1/16*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a*e*h-
7/256*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*e*h
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((h*x+g)*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 4.75911, size = 4232, normalized size = 10.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((h*x+g)*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")
[Out]
[1/430080*(105*(2*(24*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 12*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e + (
7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*f)*g - (24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - 2
*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*e + 3*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b
*c^3)*f)*h)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*
(15360*c^7*f*h*x^6 + 1280*(14*c^7*f*g + (14*c^7*e + 15*b*c^6*f)*h)*x^5 + 128*(14*(12*c^7*e + 13*b*c^6*f)*g + (
168*c^7*d + 182*b*c^6*e + 3*(b^2*c^5 + 64*a*c^6)*f)*h)*x^4 + 16*(14*(120*c^7*d + 132*b*c^6*e + (3*b^2*c^5 + 14
0*a*c^6)*f)*g + (1848*b*c^6*d + 14*(3*b^2*c^5 + 140*a*c^6)*e - 3*(9*b^3*c^4 - 44*a*b*c^5)*f)*h)*x^3 + 8*(14*(3
60*b*c^6*d + 12*(b^2*c^5 + 32*a*c^6)*e - (7*b^3*c^4 - 36*a*b*c^5)*f)*g + (168*(b^2*c^5 + 32*a*c^6)*d - 14*(7*b
^3*c^4 - 36*a*b*c^5)*e + 3*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*f)*h)*x^2 - 14*(120*(3*b^3*c^4 - 20*a*b*
c^5)*d - 12*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*e + (105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*f)*g
+ (168*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d - 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*e + 3
*(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*f)*h + 2*(14*(120*(b^2*c^5 + 20*a*c^6)*d - 12*
(5*b^3*c^4 - 28*a*b*c^5)*e + (35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*f)*g - (168*(5*b^3*c^4 - 28*a*b*c^5)*d
- 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*e + 3*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*f)*h)*x)
*sqrt(c*x^2 + b*x + a))/c^6, -1/215040*(105*(2*(24*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 12*(b^5*c^2 - 8*a*
b^3*c^3 + 16*a^2*b*c^4)*e + (7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*f)*g - (24*(b^5*c^2 - 8*a*
b^3*c^3 + 16*a^2*b*c^4)*d - 2*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*e + 3*(3*b^7 - 28*a*b^5*
c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*f)*h)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x
^2 + b*c*x + a*c)) - 2*(15360*c^7*f*h*x^6 + 1280*(14*c^7*f*g + (14*c^7*e + 15*b*c^6*f)*h)*x^5 + 128*(14*(12*c^
7*e + 13*b*c^6*f)*g + (168*c^7*d + 182*b*c^6*e + 3*(b^2*c^5 + 64*a*c^6)*f)*h)*x^4 + 16*(14*(120*c^7*d + 132*b*
c^6*e + (3*b^2*c^5 + 140*a*c^6)*f)*g + (1848*b*c^6*d + 14*(3*b^2*c^5 + 140*a*c^6)*e - 3*(9*b^3*c^4 - 44*a*b*c^
5)*f)*h)*x^3 + 8*(14*(360*b*c^6*d + 12*(b^2*c^5 + 32*a*c^6)*e - (7*b^3*c^4 - 36*a*b*c^5)*f)*g + (168*(b^2*c^5
+ 32*a*c^6)*d - 14*(7*b^3*c^4 - 36*a*b*c^5)*e + 3*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*f)*h)*x^2 - 14*(1
20*(3*b^3*c^4 - 20*a*b*c^5)*d - 12*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*e + (105*b^5*c^2 - 760*a*b^3*c^3
+ 1296*a^2*b*c^4)*f)*g + (168*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d - 14*(105*b^5*c^2 - 760*a*b^3*c^3
+ 1296*a^2*b*c^4)*e + 3*(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*f)*h + 2*(14*(120*(b^2*
c^5 + 20*a*c^6)*d - 12*(5*b^3*c^4 - 28*a*b*c^5)*e + (35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*f)*g - (168*(5*
b^3*c^4 - 28*a*b*c^5)*d - 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*e + 3*(105*b^5*c^2 - 728*a*b^3*c^3 + 1
168*a^2*b*c^4)*f)*h)*x)*sqrt(c*x^2 + b*x + a))/c^6]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g + h x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((h*x+g)*(c*x**2+b*x+a)**(3/2)*(f*x**2+e*x+d),x)
[Out]
Integral((g + h*x)*(a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2), x)
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Giac [B] time = 1.22256, size = 1289, normalized size = 3.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((h*x+g)*(c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")
[Out]
1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*c*f*h*x + (14*c^7*f*g + 15*b*c^6*f*h + 14*c^7*h*e)/c^6)*x +
(182*b*c^6*f*g + 168*c^7*d*h + 3*b^2*c^5*f*h + 192*a*c^6*f*h + 168*c^7*g*e + 182*b*c^6*h*e)/c^6)*x + (1680*c^
7*d*g + 42*b^2*c^5*f*g + 1960*a*c^6*f*g + 1848*b*c^6*d*h - 27*b^3*c^4*f*h + 132*a*b*c^5*f*h + 1848*b*c^6*g*e +
42*b^2*c^5*h*e + 1960*a*c^6*h*e)/c^6)*x + (5040*b*c^6*d*g - 98*b^3*c^4*f*g + 504*a*b*c^5*f*g + 168*b^2*c^5*d*
h + 5376*a*c^6*d*h + 63*b^4*c^3*f*h - 372*a*b^2*c^4*f*h + 384*a^2*c^5*f*h + 168*b^2*c^5*g*e + 5376*a*c^6*g*e -
98*b^3*c^4*h*e + 504*a*b*c^5*h*e)/c^6)*x + (1680*b^2*c^5*d*g + 33600*a*c^6*d*g + 490*b^4*c^3*f*g - 3024*a*b^2
*c^4*f*g + 3360*a^2*c^5*f*g - 840*b^3*c^4*d*h + 4704*a*b*c^5*d*h - 315*b^5*c^2*f*h + 2184*a*b^3*c^3*f*h - 3504
*a^2*b*c^4*f*h - 840*b^3*c^4*g*e + 4704*a*b*c^5*g*e + 490*b^4*c^3*h*e - 3024*a*b^2*c^4*h*e + 3360*a^2*c^5*h*e)
/c^6)*x - (5040*b^3*c^4*d*g - 33600*a*b*c^5*d*g + 1470*b^5*c^2*f*g - 10640*a*b^3*c^3*f*g + 18144*a^2*b*c^4*f*g
- 2520*b^4*c^3*d*h + 16800*a*b^2*c^4*d*h - 21504*a^2*c^5*d*h - 945*b^6*c*f*h + 7560*a*b^4*c^2*f*h - 16464*a^2
*b^2*c^3*f*h + 6144*a^3*c^4*f*h - 2520*b^4*c^3*g*e + 16800*a*b^2*c^4*g*e - 21504*a^2*c^5*g*e + 1470*b^5*c^2*h*
e - 10640*a*b^3*c^3*h*e + 18144*a^2*b*c^4*h*e)/c^6) - 1/2048*(48*b^4*c^3*d*g - 384*a*b^2*c^4*d*g + 768*a^2*c^5
*d*g + 14*b^6*c*f*g - 120*a*b^4*c^2*f*g + 288*a^2*b^2*c^3*f*g - 128*a^3*c^4*f*g - 24*b^5*c^2*d*h + 192*a*b^3*c
^3*d*h - 384*a^2*b*c^4*d*h - 9*b^7*f*h + 84*a*b^5*c*f*h - 240*a^2*b^3*c^2*f*h + 192*a^3*b*c^3*f*h - 24*b^5*c^2
*g*e + 192*a*b^3*c^3*g*e - 384*a^2*b*c^4*g*e + 14*b^6*c*h*e - 120*a*b^4*c^2*h*e + 288*a^2*b^2*c^3*h*e - 128*a^
3*c^4*h*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)