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

Optimal. Leaf size=462 \[ \frac{\left (a+c x^2\right )^{5/2} \left (4 \left (32 a^2 f h^4-8 a c h^2 \left (9 h (d h+3 e g)+17 f g^2\right )-3 c^2 g^2 \left (5 f g^2-3 h (64 d h+3 e g)\right )\right )-5 c h x \left (a h^2 (63 e h+61 f g)+2 c g \left (5 f g^2-9 h (12 d h+e g)\right )\right )\right )}{5040 c^3 h}+\frac{x \left (a+c x^2\right )^{3/2} \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{128 c^2}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{128 c^{5/2}}+\frac{\left (a+c x^2\right )^{5/2} (g+h x)^2 \left (8 h^2 (9 c d-4 a f)-3 c g (5 f g-9 e h)\right )}{504 c^2 h}-\frac{\left (a+c x^2\right )^{5/2} (g+h x)^3 (5 f g-9 e h)}{72 c h}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^4}{9 c h} \]

[Out]

(a*(48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x*Sqrt[a + c*x^2])/(128*c^2) +
((48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x*(a + c*x^2)^(3/2))/(192*c^2)
+ ((8*(9*c*d - 4*a*f)*h^2 - 3*c*g*(5*f*g - 9*e*h))*(g + h*x)^2*(a + c*x^2)^(5/2))/(504*c^2*h) - ((5*f*g - 9*e*
h)*(g + h*x)^3*(a + c*x^2)^(5/2))/(72*c*h) + (f*(g + h*x)^4*(a + c*x^2)^(5/2))/(9*c*h) + ((4*(32*a^2*f*h^4 - 8
*a*c*h^2*(17*f*g^2 + 9*h*(3*e*g + d*h)) - 3*c^2*g^2*(5*f*g^2 - 3*h*(3*e*g + 64*d*h))) - 5*c*h*(a*h^2*(61*f*g +
63*e*h) + 2*c*g*(5*f*g^2 - 9*h*(e*g + 12*d*h)))*x)*(a + c*x^2)^(5/2))/(5040*c^3*h) + (a^2*(48*c^2*d*g^3 + 3*a
^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.13443, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{\left (a+c x^2\right )^{5/2} \left (4 \left (32 a^2 f h^4-8 a c h^2 \left (9 h (d h+3 e g)+17 f g^2\right )-c^2 \left (15 f g^4-9 g^2 h (64 d h+3 e g)\right )\right )-5 c h x \left (a h^2 (63 e h+61 f g)+2 c \left (5 f g^3-9 g h (12 d h+e g)\right )\right )\right )}{5040 c^3 h}+\frac{x \left (a+c x^2\right )^{3/2} \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{128 c^2}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{128 c^{5/2}}+\frac{\left (a+c x^2\right )^{5/2} (g+h x)^2 \left (8 h^2 (9 c d-4 a f)-3 c g (5 f g-9 e h)\right )}{504 c^2 h}-\frac{\left (a+c x^2\right )^{5/2} (g+h x)^3 (5 f g-9 e h)}{72 c h}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^4}{9 c h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x*Sqrt[a + c*x^2])/(128*c^2) +
((48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x*(a + c*x^2)^(3/2))/(192*c^2)
+ ((8*(9*c*d - 4*a*f)*h^2 - 3*c*g*(5*f*g - 9*e*h))*(g + h*x)^2*(a + c*x^2)^(5/2))/(504*c^2*h) - ((5*f*g - 9*e*
h)*(g + h*x)^3*(a + c*x^2)^(5/2))/(72*c*h) + (f*(g + h*x)^4*(a + c*x^2)^(5/2))/(9*c*h) + ((4*(32*a^2*f*h^4 - 8
*a*c*h^2*(17*f*g^2 + 9*h*(3*e*g + d*h)) - c^2*(15*f*g^4 - 9*g^2*h*(3*e*g + 64*d*h))) - 5*c*h*(a*h^2*(61*f*g +
63*e*h) + 2*c*(5*f*g^3 - 9*g*h*(e*g + 12*d*h)))*x)*(a + c*x^2)^(5/2))/(5040*c^3*h) + (a^2*(48*c^2*d*g^3 + 3*a^
2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(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)^3 \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\int (g+h x)^3 \left ((9 c d-4 a f) h^2-c h (5 f g-9 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{9 c h^2}\\ &=-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\int (g+h x)^2 \left (c h^2 (72 c d g-17 a f g-27 a e h)+c h \left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{72 c^2 h^2}\\ &=\frac{\left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{504 c^2 h}-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\int (g+h x) \left (c h^2 \left (504 c^2 d g^2+64 a^2 f h^2-a c \left (89 f g^2+9 h (27 e g+16 d h)\right )\right )-3 c^2 h \left (a h^2 (61 f g+63 e h)+2 c \left (5 f g^3-9 g h (e g+12 d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{504 c^3 h^2}\\ &=\frac{\left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{504 c^2 h}-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\left (4 \left (32 a^2 f h^4-8 a c h^2 \left (17 f g^2+9 h (3 e g+d h)\right )-c^2 \left (15 f g^4-9 g^2 h (3 e g+64 d h)\right )\right )-5 c h \left (a h^2 (61 f g+63 e h)+2 c \left (5 f g^3-9 g h (e g+12 d h)\right )\right ) x\right ) \left (a+c x^2\right )^{5/2}}{5040 c^3 h}+\frac{\left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (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^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}+\frac{\left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{504 c^2 h}-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\left (4 \left (32 a^2 f h^4-8 a c h^2 \left (17 f g^2+9 h (3 e g+d h)\right )-c^2 \left (15 f g^4-9 g^2 h (3 e g+64 d h)\right )\right )-5 c h \left (a h^2 (61 f g+63 e h)+2 c \left (5 f g^3-9 g h (e g+12 d h)\right )\right ) x\right ) \left (a+c x^2\right )^{5/2}}{5040 c^3 h}+\frac{\left (a \left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (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^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}+\frac{\left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{504 c^2 h}-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\left (4 \left (32 a^2 f h^4-8 a c h^2 \left (17 f g^2+9 h (3 e g+d h)\right )-c^2 \left (15 f g^4-9 g^2 h (3 e g+64 d h)\right )\right )-5 c h \left (a h^2 (61 f g+63 e h)+2 c \left (5 f g^3-9 g h (e g+12 d h)\right )\right ) x\right ) \left (a+c x^2\right )^{5/2}}{5040 c^3 h}+\frac{\left (a^2 \left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 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}{128 c^2}\\ &=\frac{a \left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}+\frac{\left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{504 c^2 h}-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\left (4 \left (32 a^2 f h^4-8 a c h^2 \left (17 f g^2+9 h (3 e g+d h)\right )-c^2 \left (15 f g^4-9 g^2 h (3 e g+64 d h)\right )\right )-5 c h \left (a h^2 (61 f g+63 e h)+2 c \left (5 f g^3-9 g h (e g+12 d h)\right )\right ) x\right ) \left (a+c x^2\right )^{5/2}}{5040 c^3 h}+\frac{\left (a^2 \left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 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 )}{128 c^2}\\ &=\frac{a \left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \sqrt{a+c x^2}}{128 c^2}+\frac{\left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 a c g \left (f g^2+3 h (e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}+\frac{\left (8 (9 c d-4 a f) h^2-3 c g (5 f g-9 e h)\right ) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{504 c^2 h}-\frac{(5 f g-9 e h) (g+h x)^3 \left (a+c x^2\right )^{5/2}}{72 c h}+\frac{f (g+h x)^4 \left (a+c x^2\right )^{5/2}}{9 c h}+\frac{\left (4 \left (32 a^2 f h^4-8 a c h^2 \left (17 f g^2+9 h (3 e g+d h)\right )-c^2 \left (15 f g^4-9 g^2 h (3 e g+64 d h)\right )\right )-5 c h \left (a h^2 (61 f g+63 e h)+2 c \left (5 f g^3-9 g h (e g+12 d h)\right )\right ) x\right ) \left (a+c x^2\right )^{5/2}}{5040 c^3 h}+\frac{a^2 \left (48 c^2 d g^3+3 a^2 h^2 (3 f g+e h)-8 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 )}{128 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.573367, size = 481, normalized size = 1.04 \[ \frac{\sqrt{a+c x^2} \left (384 c^2 x^4 \left (a^2 f h^3+24 a c h \left (h (d h+3 e g)+3 f g^2\right )+21 c^2 g^2 (3 d h+e g)\right )+210 c^2 x^3 \left (3 a^2 h^2 (e h+3 f g)+56 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )+128 a c x^2 \left (-4 a^2 f h^3+9 a c h \left (h (d h+3 e g)+3 f g^2\right )+126 c^2 g^2 (3 d h+e g)\right )+315 a c x \left (-3 a^2 h^2 (e h+3 f g)+8 a c g \left (3 h (d h+e g)+f g^2\right )+80 c^2 d g^3\right )+128 a^2 \left (8 a^2 f h^3-18 a c h \left (h (d h+3 e g)+3 f g^2\right )+63 c^2 g^2 (3 d h+e g)\right )+640 c^3 h x^6 \left (10 a f h^2+9 c \left (h (d h+3 e g)+3 f g^2\right )\right )+840 c^3 x^5 \left (9 a h^2 (e h+3 f g)+8 c \left (3 g h (d h+e g)+f g^3\right )\right )+5040 c^4 h^2 x^7 (e h+3 f g)+4480 c^4 f h^3 x^8\right )+315 a^2 \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \left (3 a^2 h^2 (e h+3 f g)-8 a c g \left (3 h (d h+e g)+f g^2\right )+48 c^2 d g^3\right )}{40320 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(128*a^2*(8*a^2*f*h^3 + 63*c^2*g^2*(e*g + 3*d*h) - 18*a*c*h*(3*f*g^2 + h*(3*e*g + d*h))) + 31
5*a*c*(80*c^2*d*g^3 - 3*a^2*h^2*(3*f*g + e*h) + 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x + 128*a*c*(-4*a^2*f*h^3 +
126*c^2*g^2*(e*g + 3*d*h) + 9*a*c*h*(3*f*g^2 + h*(3*e*g + d*h)))*x^2 + 210*c^2*(48*c^2*d*g^3 + 3*a^2*h^2*(3*f
*g + e*h) + 56*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x^3 + 384*c^2*(a^2*f*h^3 + 21*c^2*g^2*(e*g + 3*d*h) + 24*a*c*h
*(3*f*g^2 + h*(3*e*g + d*h)))*x^4 + 840*c^3*(9*a*h^2*(3*f*g + e*h) + 8*c*(f*g^3 + 3*g*h*(e*g + d*h)))*x^5 + 64
0*c^3*h*(10*a*f*h^2 + 9*c*(3*f*g^2 + h*(3*e*g + d*h)))*x^6 + 5040*c^4*h^2*(3*f*g + e*h)*x^7 + 4480*c^4*f*h^3*x
^8) + 315*a^2*Sqrt[c]*(48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*Log[c*x + S
qrt[c]*Sqrt[a + c*x^2]])/(40320*c^3)

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Maple [A]  time = 0.061, size = 794, 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*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x)

[Out]

-1/8*a/c*x*(c*x^2+a)^(3/2)*e*g^2*h-3/16*a^2/c*x*(c*x^2+a)^(1/2)*d*g*h^2-3/16*a^2/c*x*(c*x^2+a)^(1/2)*e*g^2*h-3
/16*a/c^2*x*(c*x^2+a)^(5/2)*f*g*h^2+3/64*a^2/c^2*x*(c*x^2+a)^(3/2)*f*g*h^2-1/16*a^3/c^(3/2)*ln(x*c^(1/2)+(c*x^
2+a)^(1/2))*f*g^3+1/8*x^3*(c*x^2+a)^(5/2)/c*e*h^3+3/128*a^4/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*e*h^3+1/9*f*
h^3*x^4*(c*x^2+a)^(5/2)/c+8/315*f*h^3*a^2/c^3*(c*x^2+a)^(5/2)+3/5*(c*x^2+a)^(5/2)/c*d*g^2*h+1/7*x^2*(c*x^2+a)^
(5/2)/c*d*h^3-2/35*a/c^2*(c*x^2+a)^(5/2)*d*h^3+1/6*x*(c*x^2+a)^(5/2)/c*f*g^3+3/8*d*g^3*a*x*(c*x^2+a)^(1/2)+3/8
*d*g^3*a^2/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+1/5*(c*x^2+a)^(5/2)/c*e*g^3+1/4*d*g^3*x*(c*x^2+a)^(3/2)+9/128
*a^3/c^2*x*(c*x^2+a)^(1/2)*f*g*h^2-1/8*a/c*x*(c*x^2+a)^(3/2)*d*g*h^2-4/63*f*h^3*a/c^2*x^2*(c*x^2+a)^(5/2)+3/8*
x^3*(c*x^2+a)^(5/2)/c*f*g*h^2-1/16*a/c^2*x*(c*x^2+a)^(5/2)*e*h^3+1/64*a^2/c^2*x*(c*x^2+a)^(3/2)*e*h^3+3/128*a^
3/c^2*x*(c*x^2+a)^(1/2)*e*h^3+9/128*a^4/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*f*g*h^2+3/7*x^2*(c*x^2+a)^(5/2)/
c*f*g^2*h-6/35*a/c^2*(c*x^2+a)^(5/2)*e*g*h^2-6/35*a/c^2*(c*x^2+a)^(5/2)*f*g^2*h+1/2*x*(c*x^2+a)^(5/2)/c*d*g*h^
2+1/2*x*(c*x^2+a)^(5/2)/c*e*g^2*h-1/24*a/c*x*(c*x^2+a)^(3/2)*f*g^3-1/16*a^2/c*x*(c*x^2+a)^(1/2)*f*g^3-3/16*a^3
/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*d*g*h^2-3/16*a^3/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))*e*g^2*h+3/7*x^2*
(c*x^2+a)^(5/2)/c*e*g*h^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)^3*(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 = 2.57994, size = 2634, normalized size = 5.7 \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*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

[-1/80640*(315*(24*a^3*c*e*g^2*h - 3*a^4*e*h^3 - 8*(6*a^2*c^2*d - a^3*c*f)*g^3 + 3*(8*a^3*c*d - 3*a^4*f)*g*h^2
)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(4480*c^4*f*h^3*x^8 + 8064*a^2*c^2*e*g^3 - 6912*
a^3*c*e*g*h^2 + 5040*(3*c^4*f*g*h^2 + c^4*e*h^3)*x^7 + 640*(27*c^4*f*g^2*h + 27*c^4*e*g*h^2 + (9*c^4*d + 10*a*
c^3*f)*h^3)*x^6 + 840*(8*c^4*f*g^3 + 24*c^4*e*g^2*h + 9*a*c^3*e*h^3 + 3*(8*c^4*d + 9*a*c^3*f)*g*h^2)*x^5 + 384
*(21*c^4*e*g^3 + 72*a*c^3*e*g*h^2 + 9*(7*c^4*d + 8*a*c^3*f)*g^2*h + (24*a*c^3*d + a^2*c^2*f)*h^3)*x^4 + 3456*(
7*a^2*c^2*d - 2*a^3*c*f)*g^2*h - 256*(9*a^3*c*d - 4*a^4*f)*h^3 + 210*(168*a*c^3*e*g^2*h + 3*a^2*c^2*e*h^3 + 8*
(6*c^4*d + 7*a*c^3*f)*g^3 + 3*(56*a*c^3*d + 3*a^2*c^2*f)*g*h^2)*x^3 + 128*(126*a*c^3*e*g^3 + 27*a^2*c^2*e*g*h^
2 + 27*(14*a*c^3*d + a^2*c^2*f)*g^2*h + (9*a^2*c^2*d - 4*a^3*c*f)*h^3)*x^2 + 315*(24*a^2*c^2*e*g^2*h - 3*a^3*c
*e*h^3 + 8*(10*a*c^3*d + a^2*c^2*f)*g^3 + 3*(8*a^2*c^2*d - 3*a^3*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/c^3, 1/40320*
(315*(24*a^3*c*e*g^2*h - 3*a^4*e*h^3 - 8*(6*a^2*c^2*d - a^3*c*f)*g^3 + 3*(8*a^3*c*d - 3*a^4*f)*g*h^2)*sqrt(-c)
*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (4480*c^4*f*h^3*x^8 + 8064*a^2*c^2*e*g^3 - 6912*a^3*c*e*g*h^2 + 5040*(3*
c^4*f*g*h^2 + c^4*e*h^3)*x^7 + 640*(27*c^4*f*g^2*h + 27*c^4*e*g*h^2 + (9*c^4*d + 10*a*c^3*f)*h^3)*x^6 + 840*(8
*c^4*f*g^3 + 24*c^4*e*g^2*h + 9*a*c^3*e*h^3 + 3*(8*c^4*d + 9*a*c^3*f)*g*h^2)*x^5 + 384*(21*c^4*e*g^3 + 72*a*c^
3*e*g*h^2 + 9*(7*c^4*d + 8*a*c^3*f)*g^2*h + (24*a*c^3*d + a^2*c^2*f)*h^3)*x^4 + 3456*(7*a^2*c^2*d - 2*a^3*c*f)
*g^2*h - 256*(9*a^3*c*d - 4*a^4*f)*h^3 + 210*(168*a*c^3*e*g^2*h + 3*a^2*c^2*e*h^3 + 8*(6*c^4*d + 7*a*c^3*f)*g^
3 + 3*(56*a*c^3*d + 3*a^2*c^2*f)*g*h^2)*x^3 + 128*(126*a*c^3*e*g^3 + 27*a^2*c^2*e*g*h^2 + 27*(14*a*c^3*d + a^2
*c^2*f)*g^2*h + (9*a^2*c^2*d - 4*a^3*c*f)*h^3)*x^2 + 315*(24*a^2*c^2*e*g^2*h - 3*a^3*c*e*h^3 + 8*(10*a*c^3*d +
a^2*c^2*f)*g^3 + 3*(8*a^2*c^2*d - 3*a^3*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]  time = 67.6183, size = 1916, normalized size = 4.15 \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*(c*x**2+a)**(3/2)*(f*x**2+e*x+d),x)

[Out]

-3*a**(7/2)*e*h**3*x/(128*c**2*sqrt(1 + c*x**2/a)) - 9*a**(7/2)*f*g*h**2*x/(128*c**2*sqrt(1 + c*x**2/a)) + 3*a
**(5/2)*d*g*h**2*x/(16*c*sqrt(1 + c*x**2/a)) + 3*a**(5/2)*e*g**2*h*x/(16*c*sqrt(1 + c*x**2/a)) - a**(5/2)*e*h*
*3*x**3/(128*c*sqrt(1 + c*x**2/a)) + a**(5/2)*f*g**3*x/(16*c*sqrt(1 + c*x**2/a)) - 3*a**(5/2)*f*g*h**2*x**3/(1
28*c*sqrt(1 + c*x**2/a)) + a**(3/2)*d*g**3*x*sqrt(1 + c*x**2/a)/2 + a**(3/2)*d*g**3*x/(8*sqrt(1 + c*x**2/a)) +
17*a**(3/2)*d*g*h**2*x**3/(16*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*e*g**2*h*x**3/(16*sqrt(1 + c*x**2/a)) + 13*a*
*(3/2)*e*h**3*x**5/(64*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*f*g**3*x**3/(48*sqrt(1 + c*x**2/a)) + 39*a**(3/2)*f*g
*h**2*x**5/(64*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*c*d*g**3*x**3/(8*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*d*g*h**2*x*
*5/(8*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*e*g**2*h*x**5/(8*sqrt(1 + c*x**2/a)) + 5*sqrt(a)*c*e*h**3*x**7/(16*sq
rt(1 + c*x**2/a)) + 11*sqrt(a)*c*f*g**3*x**5/(24*sqrt(1 + c*x**2/a)) + 15*sqrt(a)*c*f*g*h**2*x**7/(16*sqrt(1 +
c*x**2/a)) + 3*a**4*e*h**3*asinh(sqrt(c)*x/sqrt(a))/(128*c**(5/2)) + 9*a**4*f*g*h**2*asinh(sqrt(c)*x/sqrt(a))
/(128*c**(5/2)) - 3*a**3*d*g*h**2*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2)) - 3*a**3*e*g**2*h*asinh(sqrt(c)*x/sqr
t(a))/(16*c**(3/2)) - a**3*f*g**3*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2)) + 3*a**2*d*g**3*asinh(sqrt(c)*x/sqrt(
a))/(8*sqrt(c)) + 3*a*d*g**2*h*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*d*
h**3*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)) + a*e*g**3*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c
), True)) + 3*a*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*a*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, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + a*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, True)) + 3*c*d*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, Ne(c, 0)), (sqrt(a)*x**
4/4, True)) + c*d*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, True)) + c*e*g**3*Pie
cewise((-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*c*e*g*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, T
rue)) + 3*c*f*g**2*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, True)) + c*f*h**3*Piece
wise((-16*a**4*sqrt(a + c*x**2)/(315*c**4) + 8*a**3*x**2*sqrt(a + c*x**2)/(315*c**3) - 2*a**2*x**4*sqrt(a + c*
x**2)/(105*c**2) + a*x**6*sqrt(a + c*x**2)/(63*c) + x**8*sqrt(a + c*x**2)/9, Ne(c, 0)), (sqrt(a)*x**8/8, True)
) + c**2*d*g**3*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*d*g*h**2*x**7/(2*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2
*e*g**2*h*x**7/(2*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*e*h**3*x**9/(8*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*f*g**3*
x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + 3*c**2*f*g*h**2*x**9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.22935, size = 880, normalized size = 1.9 \begin{align*} \frac{1}{40320} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (2 \,{\left (7 \,{\left (8 \, c f h^{3} x + \frac{9 \,{\left (3 \, c^{8} f g h^{2} + c^{8} h^{3} e\right )}}{c^{7}}\right )} x + \frac{8 \,{\left (27 \, c^{8} f g^{2} h + 9 \, c^{8} d h^{3} + 10 \, a c^{7} f h^{3} + 27 \, c^{8} g h^{2} e\right )}}{c^{7}}\right )} x + \frac{21 \,{\left (8 \, c^{8} f g^{3} + 24 \, c^{8} d g h^{2} + 27 \, a c^{7} f g h^{2} + 24 \, c^{8} g^{2} h e + 9 \, a c^{7} h^{3} e\right )}}{c^{7}}\right )} x + \frac{48 \,{\left (63 \, c^{8} d g^{2} h + 72 \, a c^{7} f g^{2} h + 24 \, a c^{7} d h^{3} + a^{2} c^{6} f h^{3} + 21 \, c^{8} g^{3} e + 72 \, a c^{7} g h^{2} e\right )}}{c^{7}}\right )} x + \frac{105 \,{\left (48 \, c^{8} d g^{3} + 56 \, a c^{7} f g^{3} + 168 \, a c^{7} d g h^{2} + 9 \, a^{2} c^{6} f g h^{2} + 168 \, a c^{7} g^{2} h e + 3 \, a^{2} c^{6} h^{3} e\right )}}{c^{7}}\right )} x + \frac{64 \,{\left (378 \, a c^{7} d g^{2} h + 27 \, a^{2} c^{6} f g^{2} h + 9 \, a^{2} c^{6} d h^{3} - 4 \, a^{3} c^{5} f h^{3} + 126 \, a c^{7} g^{3} e + 27 \, a^{2} c^{6} g h^{2} e\right )}}{c^{7}}\right )} x + \frac{315 \,{\left (80 \, a c^{7} d g^{3} + 8 \, a^{2} c^{6} f g^{3} + 24 \, a^{2} c^{6} d g h^{2} - 9 \, a^{3} c^{5} f g h^{2} + 24 \, a^{2} c^{6} g^{2} h e - 3 \, a^{3} c^{5} h^{3} e\right )}}{c^{7}}\right )} x + \frac{128 \,{\left (189 \, a^{2} c^{6} d g^{2} h - 54 \, a^{3} c^{5} f g^{2} h - 18 \, a^{3} c^{5} d h^{3} + 8 \, a^{4} c^{4} f h^{3} + 63 \, a^{2} c^{6} g^{3} e - 54 \, a^{3} c^{5} g h^{2} e\right )}}{c^{7}}\right )} - \frac{{\left (48 \, a^{2} c^{2} d g^{3} - 8 \, a^{3} c f g^{3} - 24 \, a^{3} c d g h^{2} + 9 \, a^{4} f g h^{2} - 24 \, a^{3} c g^{2} h e + 3 \, a^{4} h^{3} 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)^3*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/40320*sqrt(c*x^2 + a)*((2*((4*(5*(2*(7*(8*c*f*h^3*x + 9*(3*c^8*f*g*h^2 + c^8*h^3*e)/c^7)*x + 8*(27*c^8*f*g^2
*h + 9*c^8*d*h^3 + 10*a*c^7*f*h^3 + 27*c^8*g*h^2*e)/c^7)*x + 21*(8*c^8*f*g^3 + 24*c^8*d*g*h^2 + 27*a*c^7*f*g*h
^2 + 24*c^8*g^2*h*e + 9*a*c^7*h^3*e)/c^7)*x + 48*(63*c^8*d*g^2*h + 72*a*c^7*f*g^2*h + 24*a*c^7*d*h^3 + a^2*c^6
*f*h^3 + 21*c^8*g^3*e + 72*a*c^7*g*h^2*e)/c^7)*x + 105*(48*c^8*d*g^3 + 56*a*c^7*f*g^3 + 168*a*c^7*d*g*h^2 + 9*
a^2*c^6*f*g*h^2 + 168*a*c^7*g^2*h*e + 3*a^2*c^6*h^3*e)/c^7)*x + 64*(378*a*c^7*d*g^2*h + 27*a^2*c^6*f*g^2*h + 9
*a^2*c^6*d*h^3 - 4*a^3*c^5*f*h^3 + 126*a*c^7*g^3*e + 27*a^2*c^6*g*h^2*e)/c^7)*x + 315*(80*a*c^7*d*g^3 + 8*a^2*
c^6*f*g^3 + 24*a^2*c^6*d*g*h^2 - 9*a^3*c^5*f*g*h^2 + 24*a^2*c^6*g^2*h*e - 3*a^3*c^5*h^3*e)/c^7)*x + 128*(189*a
^2*c^6*d*g^2*h - 54*a^3*c^5*f*g^2*h - 18*a^3*c^5*d*h^3 + 8*a^4*c^4*f*h^3 + 63*a^2*c^6*g^3*e - 54*a^3*c^5*g*h^2
*e)/c^7) - 1/128*(48*a^2*c^2*d*g^3 - 8*a^3*c*f*g^3 - 24*a^3*c*d*g*h^2 + 9*a^4*f*g*h^2 - 24*a^3*c*g^2*h*e + 3*a
^4*h^3*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)