3.89 $$\int (g+h x)^2 (a+c x^2)^{3/2} (d+e x+f x^2) \, dx$$

Optimal. Leaf size=346 $\frac{x \left (a+c x^2\right )^{3/2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^2}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^{5/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (12 \left (8 a h^2 (e h+2 f g)+c g \left (5 f g^2-8 h (7 d h+e g)\right )\right )-5 h x \left (7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right )\right )}{1680 c^2 h}-\frac{\left (a+c x^2\right )^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^3}{8 c h}$

[Out]

(a*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*x*Sqrt[a + c*x^2])/(128*c^2) + ((48*c^2*d*g^
2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*x*(a + c*x^2)^(3/2))/(192*c^2) - ((5*f*g - 8*e*h)*(g + h*x)
^2*(a + c*x^2)^(5/2))/(56*c*h) + (f*(g + h*x)^3*(a + c*x^2)^(5/2))/(8*c*h) - ((12*(8*a*h^2*(2*f*g + e*h) + c*g
*(5*f*g^2 - 8*h*(e*g + 7*d*h))) - 5*h*(7*(8*c*d - 3*a*f)*h^2 - 2*c*g*(5*f*g - 8*e*h))*x)*(a + c*x^2)^(5/2))/(1
680*c^2*h) + (a^2*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(128*c^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.524336, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.207, Rules used = {1654, 833, 780, 195, 217, 206} $\frac{x \left (a+c x^2\right )^{3/2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^2}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^{5/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (12 \left (8 a h^2 (e h+2 f g)-8 c g h (7 d h+e g)+5 c f g^3\right )-5 h x \left (7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right )\right )}{1680 c^2 h}-\frac{\left (a+c x^2\right )^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^3}{8 c h}$

Antiderivative was successfully veriﬁed.

[In]

Int[(g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(a*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*x*Sqrt[a + c*x^2])/(128*c^2) + ((48*c^2*d*g^
2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*x*(a + c*x^2)^(3/2))/(192*c^2) - ((5*f*g - 8*e*h)*(g + h*x)
^2*(a + c*x^2)^(5/2))/(56*c*h) + (f*(g + h*x)^3*(a + c*x^2)^(5/2))/(8*c*h) - ((12*(5*c*f*g^3 - 8*c*g*h*(e*g +
7*d*h) + 8*a*h^2*(2*f*g + e*h)) - 5*h*(7*(8*c*d - 3*a*f)*h^2 - 2*c*g*(5*f*g - 8*e*h))*x)*(a + c*x^2)^(5/2))/(1
680*c^2*h) + (a^2*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(128*c^(5/2))

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (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 + 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 + 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)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 833

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

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(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 195

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

Rule 217

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 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)^2 \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac{\int (g+h x)^2 \left ((8 c d-3 a f) h^2-c h (5 f g-8 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{8 c h^2}\\ &=-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac{\int (g+h x) \left (c h^2 (56 c d g-11 a f g-16 a e h)+c h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2 h^2}\\ &=-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \sqrt{a+c x^2} \, dx}{64 c^2}\\ &=\frac{a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c^2}\\ &=\frac{a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c^2}\\ &=\frac{a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac{f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac{\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac{a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.14949, size = 346, normalized size = 1. $\frac{\sqrt{a+c x^2} \left (-\frac{280 a \left (3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\sqrt{c} x \left (5 a+2 c x^2\right ) \sqrt{\frac{c x^2}{a}+1}\right ) \left (h (d h+2 e g)+f g^2\right )}{c^{3/2} \sqrt{\frac{c x^2}{a}+1}}+\frac{105 a f h^2 \left (\frac{3 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}-\sqrt{c} x \left (3 a^2+14 a c x^2+8 c^2 x^4\right )\right )}{c^{5/2}}+1680 d g^2 \left (\frac{3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c} \sqrt{\frac{c x^2}{a}+1}}+5 a x+2 c x^3\right )+\frac{384 h \left (a+c x^2\right )^2 \left (5 c x^2-2 a\right ) (e h+2 f g)}{c^2}+\frac{2240 x \left (a+c x^2\right )^2 \left (h (d h+2 e g)+f g^2\right )}{c}+\frac{2688 g \left (a+c x^2\right )^2 (2 d h+e g)}{c}+\frac{1680 f h^2 x^3 \left (a+c x^2\right )^2}{c}\right )}{13440}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(Sqrt[a + c*x^2]*((2688*g*(e*g + 2*d*h)*(a + c*x^2)^2)/c + (2240*(f*g^2 + h*(2*e*g + d*h))*x*(a + c*x^2)^2)/c
+ (1680*f*h^2*x^3*(a + c*x^2)^2)/c + (384*h*(2*f*g + e*h)*(a + c*x^2)^2*(-2*a + 5*c*x^2))/c^2 - (280*a*(f*g^2
+ h*(2*e*g + d*h))*(Sqrt[c]*x*(5*a + 2*c*x^2)*Sqrt[1 + (c*x^2)/a] + 3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(
c^(3/2)*Sqrt[1 + (c*x^2)/a]) + (105*a*f*h^2*(-(Sqrt[c]*x*(3*a^2 + 14*a*c*x^2 + 8*c^2*x^4)) + (3*a^(5/2)*ArcSin
h[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/c^(5/2) + 1680*d*g^2*(5*a*x + 2*c*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt
[c]*x)/Sqrt[a]])/(Sqrt[c]*Sqrt[1 + (c*x^2)/a]))))/13440

________________________________________________________________________________________

Maple [A]  time = 0.057, size = 552, normalized size = 1.6 \begin{align*} -{\frac{{a}^{3}egh}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,fg{x}^{2}h}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{4\,afgh}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{af{h}^{2}x}{16\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aexgh}{12\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xegh}{8\,c}\sqrt{c{x}^{2}+a}}+{\frac{3\,d{g}^{2}{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{dx{h}^{2}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{fx{g}^{2}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{3}d{h}^{2}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{3}f{g}^{2}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,dgh}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,d{g}^{2}ax}{8}\sqrt{c{x}^{2}+a}}+{\frac{f{h}^{2}{x}^{3}}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,f{h}^{2}{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{e{x}^{2}{h}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ae{h}^{2}}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{e{g}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{d{g}^{2}x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xf{g}^{2}}{16\,c}\sqrt{c{x}^{2}+a}}+{\frac{{a}^{2}f{h}^{2}x}{64\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,f{h}^{2}{a}^{3}x}{128\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{egxh}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{axd{h}^{2}}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xd{h}^{2}}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{axf{g}^{2}}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x)

[Out]

-1/8*a^3/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*e*g*h+2/7*x^2*(c*x^2+a)^(5/2)/c*f*g*h-4/35*a/c^2*(c*x^2+a)^(5/2
)*f*g*h-1/16*f*h^2*a/c^2*x*(c*x^2+a)^(5/2)-1/12*a/c*x*(c*x^2+a)^(3/2)*e*g*h-1/8*a^2/c*x*(c*x^2+a)^(1/2)*e*g*h+
3/8*d*g^2*a^2/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+1/6*x*(c*x^2+a)^(5/2)/c*d*h^2+1/6*x*(c*x^2+a)^(5/2)/c*f*g^
2-1/16*a^3/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*d*h^2-1/16*a^3/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*f*g^2+2/
5*(c*x^2+a)^(5/2)/c*d*g*h+3/8*d*g^2*a*x*(c*x^2+a)^(1/2)+1/8*f*h^2*x^3*(c*x^2+a)^(5/2)/c+3/128*f*h^2*a^4/c^(5/2
)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+1/7*x^2*(c*x^2+a)^(5/2)/c*e*h^2-2/35*a/c^2*(c*x^2+a)^(5/2)*e*h^2+1/5*(c*x^2+a)
^(5/2)/c*e*g^2+1/4*d*g^2*x*(c*x^2+a)^(3/2)-1/16*a^2/c*x*(c*x^2+a)^(1/2)*f*g^2+1/64*f*h^2*a^2/c^2*x*(c*x^2+a)^(
3/2)+3/128*f*h^2*a^3/c^2*x*(c*x^2+a)^(1/2)+1/3*x*(c*x^2+a)^(5/2)/c*e*g*h-1/24*a/c*x*(c*x^2+a)^(3/2)*d*h^2-1/16
*a^2/c*x*(c*x^2+a)^(1/2)*d*h^2-1/24*a/c*x*(c*x^2+a)^(3/2)*f*g^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.11568, size = 1878, normalized size = 5.43 \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)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

[-1/26880*(105*(16*a^3*c*e*g*h - 8*(6*a^2*c^2*d - a^3*c*f)*g^2 + (8*a^3*c*d - 3*a^4*f)*h^2)*sqrt(c)*log(-2*c*x
^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(1680*c^4*f*h^2*x^7 + 2688*a^2*c^2*e*g^2 - 768*a^3*c*e*h^2 + 1920*(2
*c^4*f*g*h + c^4*e*h^2)*x^6 + 280*(8*c^4*f*g^2 + 16*c^4*e*g*h + (8*c^4*d + 9*a*c^3*f)*h^2)*x^5 + 384*(7*c^4*e*
g^2 + 8*a*c^3*e*h^2 + 2*(7*c^4*d + 8*a*c^3*f)*g*h)*x^4 + 70*(112*a*c^3*e*g*h + 8*(6*c^4*d + 7*a*c^3*f)*g^2 + (
56*a*c^3*d + 3*a^2*c^2*f)*h^2)*x^3 + 768*(7*a^2*c^2*d - 2*a^3*c*f)*g*h + 384*(14*a*c^3*e*g^2 + a^2*c^2*e*h^2 +
2*(14*a*c^3*d + a^2*c^2*f)*g*h)*x^2 + 105*(16*a^2*c^2*e*g*h + 8*(10*a*c^3*d + a^2*c^2*f)*g^2 + (8*a^2*c^2*d -
3*a^3*c*f)*h^2)*x)*sqrt(c*x^2 + a))/c^3, 1/13440*(105*(16*a^3*c*e*g*h - 8*(6*a^2*c^2*d - a^3*c*f)*g^2 + (8*a^
3*c*d - 3*a^4*f)*h^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (1680*c^4*f*h^2*x^7 + 2688*a^2*c^2*e*g^2 -
768*a^3*c*e*h^2 + 1920*(2*c^4*f*g*h + c^4*e*h^2)*x^6 + 280*(8*c^4*f*g^2 + 16*c^4*e*g*h + (8*c^4*d + 9*a*c^3*f
)*h^2)*x^5 + 384*(7*c^4*e*g^2 + 8*a*c^3*e*h^2 + 2*(7*c^4*d + 8*a*c^3*f)*g*h)*x^4 + 70*(112*a*c^3*e*g*h + 8*(6*
c^4*d + 7*a*c^3*f)*g^2 + (56*a*c^3*d + 3*a^2*c^2*f)*h^2)*x^3 + 768*(7*a^2*c^2*d - 2*a^3*c*f)*g*h + 384*(14*a*c
^3*e*g^2 + a^2*c^2*e*h^2 + 2*(14*a*c^3*d + a^2*c^2*f)*g*h)*x^2 + 105*(16*a^2*c^2*e*g*h + 8*(10*a*c^3*d + a^2*c
^2*f)*g^2 + (8*a^2*c^2*d - 3*a^3*c*f)*h^2)*x)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]  time = 50.0334, size = 1304, normalized size = 3.77 \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)**2*(c*x**2+a)**(3/2)*(f*x**2+e*x+d),x)

[Out]

-3*a**(7/2)*f*h**2*x/(128*c**2*sqrt(1 + c*x**2/a)) + a**(5/2)*d*h**2*x/(16*c*sqrt(1 + c*x**2/a)) + a**(5/2)*e*
g*h*x/(8*c*sqrt(1 + c*x**2/a)) + a**(5/2)*f*g**2*x/(16*c*sqrt(1 + c*x**2/a)) - a**(5/2)*f*h**2*x**3/(128*c*sqr
t(1 + c*x**2/a)) + a**(3/2)*d*g**2*x*sqrt(1 + c*x**2/a)/2 + a**(3/2)*d*g**2*x/(8*sqrt(1 + c*x**2/a)) + 17*a**(
3/2)*d*h**2*x**3/(48*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*e*g*h*x**3/(24*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*f*g**2
*x**3/(48*sqrt(1 + c*x**2/a)) + 13*a**(3/2)*f*h**2*x**5/(64*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*c*d*g**2*x**3/(8*s
qrt(1 + c*x**2/a)) + 11*sqrt(a)*c*d*h**2*x**5/(24*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*e*g*h*x**5/(12*sqrt(1 + c
*x**2/a)) + 11*sqrt(a)*c*f*g**2*x**5/(24*sqrt(1 + c*x**2/a)) + 5*sqrt(a)*c*f*h**2*x**7/(16*sqrt(1 + c*x**2/a))
+ 3*a**4*f*h**2*asinh(sqrt(c)*x/sqrt(a))/(128*c**(5/2)) - a**3*d*h**2*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2))
- a**3*e*g*h*asinh(sqrt(c)*x/sqrt(a))/(8*c**(3/2)) - a**3*f*g**2*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2)) + 3*a*
*2*d*g**2*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) + 2*a*d*g*h*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)
**(3/2)/(3*c), True)) + a*e*g**2*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*
e*h**2*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/
5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 2*a*f*g*h*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(
a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 2*c*d*g*h*Piecewise((-2*a**
2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x
**4/4, True)) + c*e*g**2*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4
*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + c*e*h**2*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**
3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c,
0)), (sqrt(a)*x**6/6, True)) + 2*c*f*g*h*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a +
c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, Tr
ue)) + c**2*d*g**2*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*d*h**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c**
2*e*g*h*x**7/(3*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*f*g**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*f*h**2*x*
*9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.18857, size = 610, normalized size = 1.76 \begin{align*} \frac{1}{13440} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \,{\left (7 \, c f h^{2} x + \frac{8 \,{\left (2 \, c^{7} f g h + c^{7} h^{2} e\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (8 \, c^{7} f g^{2} + 8 \, c^{7} d h^{2} + 9 \, a c^{6} f h^{2} + 16 \, c^{7} g h e\right )}}{c^{6}}\right )} x + \frac{48 \,{\left (14 \, c^{7} d g h + 16 \, a c^{6} f g h + 7 \, c^{7} g^{2} e + 8 \, a c^{6} h^{2} e\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (48 \, c^{7} d g^{2} + 56 \, a c^{6} f g^{2} + 56 \, a c^{6} d h^{2} + 3 \, a^{2} c^{5} f h^{2} + 112 \, a c^{6} g h e\right )}}{c^{6}}\right )} x + \frac{192 \,{\left (28 \, a c^{6} d g h + 2 \, a^{2} c^{5} f g h + 14 \, a c^{6} g^{2} e + a^{2} c^{5} h^{2} e\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (80 \, a c^{6} d g^{2} + 8 \, a^{2} c^{5} f g^{2} + 8 \, a^{2} c^{5} d h^{2} - 3 \, a^{3} c^{4} f h^{2} + 16 \, a^{2} c^{5} g h e\right )}}{c^{6}}\right )} x + \frac{384 \,{\left (14 \, a^{2} c^{5} d g h - 4 \, a^{3} c^{4} f g h + 7 \, a^{2} c^{5} g^{2} e - 2 \, a^{3} c^{4} h^{2} e\right )}}{c^{6}}\right )} - \frac{{\left (48 \, a^{2} c^{2} d g^{2} - 8 \, a^{3} c f g^{2} - 8 \, a^{3} c d h^{2} + 3 \, a^{4} f h^{2} - 16 \, a^{3} c g h e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/13440*sqrt(c*x^2 + a)*((2*((4*(5*(6*(7*c*f*h^2*x + 8*(2*c^7*f*g*h + c^7*h^2*e)/c^6)*x + 7*(8*c^7*f*g^2 + 8*c
^7*d*h^2 + 9*a*c^6*f*h^2 + 16*c^7*g*h*e)/c^6)*x + 48*(14*c^7*d*g*h + 16*a*c^6*f*g*h + 7*c^7*g^2*e + 8*a*c^6*h^
2*e)/c^6)*x + 35*(48*c^7*d*g^2 + 56*a*c^6*f*g^2 + 56*a*c^6*d*h^2 + 3*a^2*c^5*f*h^2 + 112*a*c^6*g*h*e)/c^6)*x +
192*(28*a*c^6*d*g*h + 2*a^2*c^5*f*g*h + 14*a*c^6*g^2*e + a^2*c^5*h^2*e)/c^6)*x + 105*(80*a*c^6*d*g^2 + 8*a^2*
c^5*f*g^2 + 8*a^2*c^5*d*h^2 - 3*a^3*c^4*f*h^2 + 16*a^2*c^5*g*h*e)/c^6)*x + 384*(14*a^2*c^5*d*g*h - 4*a^3*c^4*f
*g*h + 7*a^2*c^5*g^2*e - 2*a^3*c^4*h^2*e)/c^6) - 1/128*(48*a^2*c^2*d*g^2 - 8*a^3*c*f*g^2 - 8*a^3*c*d*h^2 + 3*a
^4*f*h^2 - 16*a^3*c*g*h*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)