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

Optimal. Leaf size=390 \[ \frac{\left (a+c x^2\right )^{3/2} \left (8 \left (8 a^2 f h^4-2 a c h^2 \left (7 h (d h+3 e g)+15 f g^2\right )-c^2 g^2 \left (3 f g^2-7 h (12 d h+e g)\right )\right )-3 c h x \left (a h^2 (35 e h+41 f g)+2 c g \left (3 f g^2-7 h (7 d h+e g)\right )\right )\right )}{840 c^3 h}+\frac{x \sqrt{a+c x^2} \left (a^2 h^2 (e h+3 f g)-2 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{16 c^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (a^2 h^2 (e h+3 f g)-2 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{16 c^{5/2}}-\frac{\left (a+c x^2\right )^{3/2} (g+h x)^2 \left (8 a f h^2+c \left (3 f g^2-7 h (2 d h+e g)\right )\right )}{70 c^2 h}-\frac{\left (a+c x^2\right )^{3/2} (g+h x)^3 (3 f g-7 e h)}{42 c h}+\frac{f \left (a+c x^2\right )^{3/2} (g+h x)^4}{7 c h} \]

[Out]

((8*c^2*d*g^3 + a^2*h^2*(3*f*g + e*h) - 2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x*Sqrt[a + c*x^2])/(16*c^2) - ((8*a
*f*h^2 + c*(3*f*g^2 - 7*h*(e*g + 2*d*h)))*(g + h*x)^2*(a + c*x^2)^(3/2))/(70*c^2*h) - ((3*f*g - 7*e*h)*(g + h*
x)^3*(a + c*x^2)^(3/2))/(42*c*h) + (f*(g + h*x)^4*(a + c*x^2)^(3/2))/(7*c*h) + ((8*(8*a^2*f*h^4 - 2*a*c*h^2*(1
5*f*g^2 + 7*h*(3*e*g + d*h)) - c^2*g^2*(3*f*g^2 - 7*h*(e*g + 12*d*h))) - 3*c*h*(a*h^2*(41*f*g + 35*e*h) + 2*c*
g*(3*f*g^2 - 7*h*(e*g + 7*d*h)))*x)*(a + c*x^2)^(3/2))/(840*c^3*h) + (a*(8*c^2*d*g^3 + a^2*h^2*(3*f*g + e*h) -
2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.829677, antiderivative size = 387, normalized size of antiderivative = 0.99, 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} \[ \frac{\left (a+c x^2\right )^{3/2} \left (8 \left (8 a^2 f h^4-2 a c h^2 \left (7 h (d h+3 e g)+15 f g^2\right )-c^2 \left (3 f g^4-7 g^2 h (12 d h+e g)\right )\right )-3 c h x \left (a h^2 (35 e h+41 f g)-14 c g h (7 d h+e g)+6 c f g^3\right )\right )}{840 c^3 h}+\frac{x \sqrt{a+c x^2} \left (a^2 h^2 (e h+3 f g)-2 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{16 c^2}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (a^2 h^2 (e h+3 f g)-2 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{16 c^{5/2}}-\frac{\left (a+c x^2\right )^{3/2} (g+h x)^2 \left (8 a f h^2-7 c h (2 d h+e g)+3 c f g^2\right )}{70 c^2 h}-\frac{\left (a+c x^2\right )^{3/2} (g+h x)^3 (3 f g-7 e h)}{42 c h}+\frac{f \left (a+c x^2\right )^{3/2} (g+h x)^4}{7 c h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*c^2*d*g^3 + a^2*h^2*(3*f*g + e*h) - 2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x*Sqrt[a + c*x^2])/(16*c^2) - ((3*c
*f*g^2 + 8*a*f*h^2 - 7*c*h*(e*g + 2*d*h))*(g + h*x)^2*(a + c*x^2)^(3/2))/(70*c^2*h) - ((3*f*g - 7*e*h)*(g + h*
x)^3*(a + c*x^2)^(3/2))/(42*c*h) + (f*(g + h*x)^4*(a + c*x^2)^(3/2))/(7*c*h) + ((8*(8*a^2*f*h^4 - 2*a*c*h^2*(1
5*f*g^2 + 7*h*(3*e*g + d*h)) - c^2*(3*f*g^4 - 7*g^2*h*(e*g + 12*d*h))) - 3*c*h*(6*c*f*g^3 - 14*c*g*h*(e*g + 7*
d*h) + a*h^2*(41*f*g + 35*e*h))*x)*(a + c*x^2)^(3/2))/(840*c^3*h) + (a*(8*c^2*d*g^3 + a^2*h^2*(3*f*g + e*h) -
2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*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)^3 \sqrt{a+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\int (g+h x)^3 \left ((7 c d-4 a f) h^2-c h (3 f g-7 e h) x\right ) \sqrt{a+c x^2} \, dx}{7 c h^2}\\ &=-\frac{(3 f g-7 e h) (g+h x)^3 \left (a+c x^2\right )^{3/2}}{42 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\int (g+h x)^2 \left (3 c h^2 (14 c d g-5 a f g-7 a e h)-3 c h \left (3 c f g^2+8 a f h^2-7 c h (e g+2 d h)\right ) x\right ) \sqrt{a+c x^2} \, dx}{42 c^2 h^2}\\ &=-\frac{\left (3 c f g^2+8 a f h^2-7 c h (e g+2 d h)\right ) (g+h x)^2 \left (a+c x^2\right )^{3/2}}{70 c^2 h}-\frac{(3 f g-7 e h) (g+h x)^3 \left (a+c x^2\right )^{3/2}}{42 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\int (g+h x) \left (3 c h^2 \left (70 c^2 d g^2+16 a^2 f h^2-a c \left (19 f g^2+7 h (7 e g+4 d h)\right )\right )-3 c^2 h \left (6 c f g^3-14 c g h (e g+7 d h)+a h^2 (41 f g+35 e h)\right ) x\right ) \sqrt{a+c x^2} \, dx}{210 c^3 h^2}\\ &=-\frac{\left (3 c f g^2+8 a f h^2-7 c h (e g+2 d h)\right ) (g+h x)^2 \left (a+c x^2\right )^{3/2}}{70 c^2 h}-\frac{(3 f g-7 e h) (g+h x)^3 \left (a+c x^2\right )^{3/2}}{42 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\left (8 \left (8 a^2 f h^4-2 a c h^2 \left (15 f g^2+7 h (3 e g+d h)\right )-c^2 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )\right )-3 c h \left (6 c f g^3-14 c g h (e g+7 d h)+a h^2 (41 f g+35 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{840 c^3 h}+\frac{\left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (e g+d h)\right )\right ) \int \sqrt{a+c x^2} \, dx}{8 c^2}\\ &=\frac{\left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \sqrt{a+c x^2}}{16 c^2}-\frac{\left (3 c f g^2+8 a f h^2-7 c h (e g+2 d h)\right ) (g+h x)^2 \left (a+c x^2\right )^{3/2}}{70 c^2 h}-\frac{(3 f g-7 e h) (g+h x)^3 \left (a+c x^2\right )^{3/2}}{42 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\left (8 \left (8 a^2 f h^4-2 a c h^2 \left (15 f g^2+7 h (3 e g+d h)\right )-c^2 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )\right )-3 c h \left (6 c f g^3-14 c g h (e g+7 d h)+a h^2 (41 f g+35 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{840 c^3 h}+\frac{\left (a \left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (e g+d h)\right )\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c^2}\\ &=\frac{\left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \sqrt{a+c x^2}}{16 c^2}-\frac{\left (3 c f g^2+8 a f h^2-7 c h (e g+2 d h)\right ) (g+h x)^2 \left (a+c x^2\right )^{3/2}}{70 c^2 h}-\frac{(3 f g-7 e h) (g+h x)^3 \left (a+c x^2\right )^{3/2}}{42 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\left (8 \left (8 a^2 f h^4-2 a c h^2 \left (15 f g^2+7 h (3 e g+d h)\right )-c^2 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )\right )-3 c h \left (6 c f g^3-14 c g h (e g+7 d h)+a h^2 (41 f g+35 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{840 c^3 h}+\frac{\left (a \left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (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 )}{16 c^2}\\ &=\frac{\left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \sqrt{a+c x^2}}{16 c^2}-\frac{\left (3 c f g^2+8 a f h^2-7 c h (e g+2 d h)\right ) (g+h x)^2 \left (a+c x^2\right )^{3/2}}{70 c^2 h}-\frac{(3 f g-7 e h) (g+h x)^3 \left (a+c x^2\right )^{3/2}}{42 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{3/2}}{7 c h}+\frac{\left (8 \left (8 a^2 f h^4-2 a c h^2 \left (15 f g^2+7 h (3 e g+d h)\right )-c^2 \left (3 f g^4-7 g^2 h (e g+12 d h)\right )\right )-3 c h \left (6 c f g^3-14 c g h (e g+7 d h)+a h^2 (41 f g+35 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{840 c^3 h}+\frac{a \left (8 c^2 d g^3+a^2 h^2 (3 f g+e h)-2 a c g \left (f g^2+3 h (e g+d h)\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.470584, size = 362, normalized size = 0.93 \[ \frac{\sqrt{a+c x^2} \left (16 c x^2 \left (-4 a^2 f h^3+7 a c h \left (h (d h+3 e g)+3 f g^2\right )+35 c^2 g^2 (3 d h+e g)\right )+105 c x \left (-a^2 h^2 (e h+3 f g)+2 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )+16 a \left (8 a^2 f h^3-14 a c h \left (h (d h+3 e g)+3 f g^2\right )+35 c^2 g^2 (3 d h+e g)\right )+48 c^2 h x^4 \left (a f h^2+7 c \left (h (d h+3 e g)+3 f g^2\right )\right )+70 c^2 x^3 \left (a h^2 (e h+3 f g)+6 c \left (3 g h (d h+e g)+f g^3\right )\right )+280 c^3 h^2 x^5 (e h+3 f g)+240 c^3 f h^3 x^6\right )+105 a \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (a^2 h^2 (e h+3 f g)-2 a c g \left (3 h (d h+e g)+f g^2\right )+8 c^2 d g^3\right )}{1680 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(16*a*(8*a^2*f*h^3 + 35*c^2*g^2*(e*g + 3*d*h) - 14*a*c*h*(3*f*g^2 + h*(3*e*g + d*h))) + 105*c
*(8*c^2*d*g^3 - a^2*h^2*(3*f*g + e*h) + 2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x + 16*c*(-4*a^2*f*h^3 + 35*c^2*g^2
*(e*g + 3*d*h) + 7*a*c*h*(3*f*g^2 + h*(3*e*g + d*h)))*x^2 + 70*c^2*(a*h^2*(3*f*g + e*h) + 6*c*(f*g^3 + 3*g*h*(
e*g + d*h)))*x^3 + 48*c^2*h*(a*f*h^2 + 7*c*(3*f*g^2 + h*(3*e*g + d*h)))*x^4 + 280*c^3*h^2*(3*f*g + e*h)*x^5 +
240*c^3*f*h^3*x^6) + 105*a*Sqrt[c]*(8*c^2*d*g^3 + a^2*h^2*(3*f*g + e*h) - 2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*L
og[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(1680*c^3)

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Maple [A]  time = 0.058, size = 661, normalized size = 1.7 \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)^3*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x)

[Out]

8/105*f*h^3*a^2/c^3*(c*x^2+a)^(3/2)+1/7*f*h^3*x^4*(c*x^2+a)^(3/2)/c+(c*x^2+a)^(3/2)/c*d*g^2*h+1/2*d*g^3*a/c^(1
/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))-2/5*a/c^2*(c*x^2+a)^(3/2)*e*g*h^2-2/5*a/c^2*(c*x^2+a)^(3/2)*f*g^2*h+3/16*a^3
/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*f*g*h^2-4/35*f*h^3*a/c^2*x^2*(c*x^2+a)^(3/2)+3/4*x*(c*x^2+a)^(3/2)/c*d*
g*h^2+3/4*x*(c*x^2+a)^(3/2)/c*e*g^2*h-1/8*a/c*x*(c*x^2+a)^(1/2)*f*g^3-3/8*a^2/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(
1/2))*d*g*h^2+1/2*d*g^3*x*(c*x^2+a)^(1/2)+1/3*(c*x^2+a)^(3/2)/c*e*g^3+3/5*x^2*(c*x^2+a)^(3/2)/c*f*g^2*h+1/16*a
^2/c^2*x*(c*x^2+a)^(1/2)*e*h^3+3/5*x^2*(c*x^2+a)^(3/2)/c*e*g*h^2-3/8*a^2/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))
*e*g^2*h-1/8*a/c^2*x*(c*x^2+a)^(3/2)*e*h^3+1/2*x^3*(c*x^2+a)^(3/2)/c*f*g*h^2+1/5*x^2*(c*x^2+a)^(3/2)/c*d*h^3-2
/15*a/c^2*(c*x^2+a)^(3/2)*d*h^3+1/16*a^3/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*e*h^3+1/6*x^3*(c*x^2+a)^(3/2)/c
*e*h^3+1/4*x*(c*x^2+a)^(3/2)/c*f*g^3-1/8*a^2/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*f*g^3-3/8*a/c^2*x*(c*x^2+a)
^(3/2)*f*g*h^2+3/16*a^2/c^2*x*(c*x^2+a)^(1/2)*f*g*h^2-3/8*a/c*x*(c*x^2+a)^(1/2)*d*g*h^2-3/8*a/c*x*(c*x^2+a)^(1
/2)*e*g^2*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)^3*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64562, size = 1889, normalized size = 4.84 \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)^3*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/3360*(105*(6*a^2*c*e*g^2*h - a^3*e*h^3 - 2*(4*a*c^2*d - a^2*c*f)*g^3 + 3*(2*a^2*c*d - a^3*f)*g*h^2)*sqrt(c
)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(240*c^3*f*h^3*x^6 + 560*a*c^2*e*g^3 - 672*a^2*c*e*g*h^2
+ 280*(3*c^3*f*g*h^2 + c^3*e*h^3)*x^5 + 48*(21*c^3*f*g^2*h + 21*c^3*e*g*h^2 + (7*c^3*d + a*c^2*f)*h^3)*x^4 +
336*(5*a*c^2*d - 2*a^2*c*f)*g^2*h - 32*(7*a^2*c*d - 4*a^3*f)*h^3 + 70*(6*c^3*f*g^3 + 18*c^3*e*g^2*h + a*c^2*e*
h^3 + 3*(6*c^3*d + a*c^2*f)*g*h^2)*x^3 + 16*(35*c^3*e*g^3 + 21*a*c^2*e*g*h^2 + 21*(5*c^3*d + a*c^2*f)*g^2*h +
(7*a*c^2*d - 4*a^2*c*f)*h^3)*x^2 + 105*(6*a*c^2*e*g^2*h - a^2*c*e*h^3 + 2*(4*c^3*d + a*c^2*f)*g^3 + 3*(2*a*c^2
*d - a^2*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/c^3, 1/1680*(105*(6*a^2*c*e*g^2*h - a^3*e*h^3 - 2*(4*a*c^2*d - a^2*c*
f)*g^3 + 3*(2*a^2*c*d - a^3*f)*g*h^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (240*c^3*f*h^3*x^6 + 560*a
*c^2*e*g^3 - 672*a^2*c*e*g*h^2 + 280*(3*c^3*f*g*h^2 + c^3*e*h^3)*x^5 + 48*(21*c^3*f*g^2*h + 21*c^3*e*g*h^2 + (
7*c^3*d + a*c^2*f)*h^3)*x^4 + 336*(5*a*c^2*d - 2*a^2*c*f)*g^2*h - 32*(7*a^2*c*d - 4*a^3*f)*h^3 + 70*(6*c^3*f*g
^3 + 18*c^3*e*g^2*h + a*c^2*e*h^3 + 3*(6*c^3*d + a*c^2*f)*g*h^2)*x^3 + 16*(35*c^3*e*g^3 + 21*a*c^2*e*g*h^2 + 2
1*(5*c^3*d + a*c^2*f)*g^2*h + (7*a*c^2*d - 4*a^2*c*f)*h^3)*x^2 + 105*(6*a*c^2*e*g^2*h - a^2*c*e*h^3 + 2*(4*c^3
*d + a*c^2*f)*g^3 + 3*(2*a*c^2*d - a^2*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]  time = 25.4176, size = 1088, normalized size = 2.79 \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)**3*(f*x**2+e*x+d)*(c*x**2+a)**(1/2),x)

[Out]

-a**(5/2)*e*h**3*x/(16*c**2*sqrt(1 + c*x**2/a)) - 3*a**(5/2)*f*g*h**2*x/(16*c**2*sqrt(1 + c*x**2/a)) + 3*a**(3
/2)*d*g*h**2*x/(8*c*sqrt(1 + c*x**2/a)) + 3*a**(3/2)*e*g**2*h*x/(8*c*sqrt(1 + c*x**2/a)) - a**(3/2)*e*h**3*x**
3/(48*c*sqrt(1 + c*x**2/a)) + a**(3/2)*f*g**3*x/(8*c*sqrt(1 + c*x**2/a)) - a**(3/2)*f*g*h**2*x**3/(16*c*sqrt(1
+ c*x**2/a)) + sqrt(a)*d*g**3*x*sqrt(1 + c*x**2/a)/2 + 9*sqrt(a)*d*g*h**2*x**3/(8*sqrt(1 + c*x**2/a)) + 9*sqr
t(a)*e*g**2*h*x**3/(8*sqrt(1 + c*x**2/a)) + 5*sqrt(a)*e*h**3*x**5/(24*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*f*g**3*x
**3/(8*sqrt(1 + c*x**2/a)) + 5*sqrt(a)*f*g*h**2*x**5/(8*sqrt(1 + c*x**2/a)) + a**3*e*h**3*asinh(sqrt(c)*x/sqrt
(a))/(16*c**(5/2)) + 3*a**3*f*g*h**2*asinh(sqrt(c)*x/sqrt(a))/(16*c**(5/2)) - 3*a**2*d*g*h**2*asinh(sqrt(c)*x/
sqrt(a))/(8*c**(3/2)) - 3*a**2*e*g**2*h*asinh(sqrt(c)*x/sqrt(a))/(8*c**(3/2)) - a**2*f*g**3*asinh(sqrt(c)*x/sq
rt(a))/(8*c**(3/2)) + a*d*g**3*asinh(sqrt(c)*x/sqrt(a))/(2*sqrt(c)) + 3*d*g**2*h*Piecewise((sqrt(a)*x**2/2, Eq
(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + d*h**3*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sq
rt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + e*g**3*Piecewise((sqrt(a
)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + 3*e*g*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)) + 3*f*g**2
*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, N
e(c, 0)), (sqrt(a)*x**4/4, True)) + f*h**3*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, T
rue)) + 3*c*d*g*h**2*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + 3*c*e*g**2*h*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) +
c*e*h**3*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c*f*g**3*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c*f*g*h**2*x**7/
(2*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.18769, size = 641, normalized size = 1.64 \begin{align*} \frac{1}{1680} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \, f h^{3} x + \frac{7 \,{\left (3 \, c^{5} f g h^{2} + c^{5} h^{3} e\right )}}{c^{5}}\right )} x + \frac{6 \,{\left (21 \, c^{5} f g^{2} h + 7 \, c^{5} d h^{3} + a c^{4} f h^{3} + 21 \, c^{5} g h^{2} e\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (6 \, c^{5} f g^{3} + 18 \, c^{5} d g h^{2} + 3 \, a c^{4} f g h^{2} + 18 \, c^{5} g^{2} h e + a c^{4} h^{3} e\right )}}{c^{5}}\right )} x + \frac{8 \,{\left (105 \, c^{5} d g^{2} h + 21 \, a c^{4} f g^{2} h + 7 \, a c^{4} d h^{3} - 4 \, a^{2} c^{3} f h^{3} + 35 \, c^{5} g^{3} e + 21 \, a c^{4} g h^{2} e\right )}}{c^{5}}\right )} x + \frac{105 \,{\left (8 \, c^{5} d g^{3} + 2 \, a c^{4} f g^{3} + 6 \, a c^{4} d g h^{2} - 3 \, a^{2} c^{3} f g h^{2} + 6 \, a c^{4} g^{2} h e - a^{2} c^{3} h^{3} e\right )}}{c^{5}}\right )} x + \frac{16 \,{\left (105 \, a c^{4} d g^{2} h - 42 \, a^{2} c^{3} f g^{2} h - 14 \, a^{2} c^{3} d h^{3} + 8 \, a^{3} c^{2} f h^{3} + 35 \, a c^{4} g^{3} e - 42 \, a^{2} c^{3} g h^{2} e\right )}}{c^{5}}\right )} - \frac{{\left (8 \, a c^{2} d g^{3} - 2 \, a^{2} c f g^{3} - 6 \, a^{2} c d g h^{2} + 3 \, a^{3} f g h^{2} - 6 \, a^{2} c g^{2} h e + a^{3} h^{3} e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^3*(f*x^2+e*x+d)*(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/1680*sqrt(c*x^2 + a)*((2*((4*(5*(6*f*h^3*x + 7*(3*c^5*f*g*h^2 + c^5*h^3*e)/c^5)*x + 6*(21*c^5*f*g^2*h + 7*c^
5*d*h^3 + a*c^4*f*h^3 + 21*c^5*g*h^2*e)/c^5)*x + 35*(6*c^5*f*g^3 + 18*c^5*d*g*h^2 + 3*a*c^4*f*g*h^2 + 18*c^5*g
^2*h*e + a*c^4*h^3*e)/c^5)*x + 8*(105*c^5*d*g^2*h + 21*a*c^4*f*g^2*h + 7*a*c^4*d*h^3 - 4*a^2*c^3*f*h^3 + 35*c^
5*g^3*e + 21*a*c^4*g*h^2*e)/c^5)*x + 105*(8*c^5*d*g^3 + 2*a*c^4*f*g^3 + 6*a*c^4*d*g*h^2 - 3*a^2*c^3*f*g*h^2 +
6*a*c^4*g^2*h*e - a^2*c^3*h^3*e)/c^5)*x + 16*(105*a*c^4*d*g^2*h - 42*a^2*c^3*f*g^2*h - 14*a^2*c^3*d*h^3 + 8*a^
3*c^2*f*h^3 + 35*a*c^4*g^3*e - 42*a^2*c^3*g*h^2*e)/c^5) - 1/16*(8*a*c^2*d*g^3 - 2*a^2*c*f*g^3 - 6*a^2*c*d*g*h^
2 + 3*a^3*f*g*h^2 - 6*a^2*c*g^2*h*e + a^3*h^3*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)