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

Optimal. Leaf size=213 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2+c \left (5 f g^2-7 h (d h+e g)\right )\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac{x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac{a x \sqrt{a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]

[Out]

(a*(6*c*d*g - a*f*g - a*e*h)*x*Sqrt[a + c*x^2])/(16*c) + ((6*c*d*g - a*(f*g + e*h))*x*(a + c*x^2)^(3/2))/(24*c
) + (f*(g + h*x)^2*(a + c*x^2)^(5/2))/(7*c*h) - ((6*(2*a*f*h^2 + c*(5*f*g^2 - 7*h*(e*g + d*h))) + 5*c*h*(5*f*g
- 7*e*h)*x)*(a + c*x^2)^(5/2))/(210*c^2*h) + (a^2*(6*c*d*g - a*f*g - a*e*h)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^
2]])/(16*c^(3/2))

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Rubi [A]  time = 0.270914, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1654, 780, 195, 217, 206} \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac{\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2-7 c h (d h+e g)+5 c f g^2\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac{x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac{a x \sqrt{a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac{f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(6*c*d*g - a*f*g - a*e*h)*x*Sqrt[a + c*x^2])/(16*c) + ((6*c*d*g - a*(f*g + e*h))*x*(a + c*x^2)^(3/2))/(24*c
) + (f*(g + h*x)^2*(a + c*x^2)^(5/2))/(7*c*h) - ((6*(5*c*f*g^2 + 2*a*f*h^2 - 7*c*h*(e*g + d*h)) + 5*c*h*(5*f*g
- 7*e*h)*x)*(a + c*x^2)^(5/2))/(210*c^2*h) + (a^2*(6*c*d*g - a*f*g - a*e*h)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^
2]])/(16*c^(3/2))

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 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) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}+\frac{\int (g+h x) \left ((7 c d-2 a f) h^2-c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c h^2}\\ &=\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{(6 c d g-a f g-a e h) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{(a (6 c d g-a f g-a e h)) \int \sqrt{a+c x^2} \, dx}{8 c}\\ &=\frac{a (6 c d g-a f g-a e h) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{\left (a^2 (6 c d g-a f g-a e h)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c}\\ &=\frac{a (6 c d g-a f g-a e h) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{\left (a^2 (6 c d g-a f g-a e h)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{16 c}\\ &=\frac{a (6 c d g-a f g-a e h) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac{\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac{a^2 (6 c d g-a f g-a e h) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.688368, size = 209, normalized size = 0.98 \[ \frac{\sqrt{a+c x^2} \left (-\frac{105 a^{5/2} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a e h+a f g-6 c d g)}{c^{3/2} \left (a+c x^2\right )}-\frac{96 a^3 f h}{c^2}+\frac{3 a^2 (112 d h+7 e (16 g+5 h x)+f x (35 g+16 h x))}{c}+2 a x (21 d (25 g+16 h x)+x (7 e (48 g+35 h x)+f x (245 g+192 h x)))+4 c x^3 (21 d (5 g+4 h x)+2 x (7 e (6 g+5 h x)+5 f x (7 g+6 h x)))\right )}{1680} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*((-96*a^3*f*h)/c^2 + (3*a^2*(112*d*h + 7*e*(16*g + 5*h*x) + f*x*(35*g + 16*h*x)))/c + 4*c*x^3
*(21*d*(5*g + 4*h*x) + 2*x*(7*e*(6*g + 5*h*x) + 5*f*x*(7*g + 6*h*x))) + 2*a*x*(21*d*(25*g + 16*h*x) + x*(7*e*(
48*g + 35*h*x) + f*x*(245*g + 192*h*x))) - (105*a^(5/2)*(-6*c*d*g + a*f*g + a*e*h)*Sqrt[1 + (c*x^2)/a]*ArcSinh
[(Sqrt[c]*x)/Sqrt[a]])/(c^(3/2)*(a + c*x^2))))/1680

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Maple [A]  time = 0.052, size = 287, normalized size = 1.4 \begin{align*}{\frac{fh{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,afh}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{ehx}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{fgx}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aehx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{afgx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}xeh}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{2}xfg}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{3}eh}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{3}fg}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{dh}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{eg}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{dgx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adgx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,dg{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*(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 = 1.47333, size = 1107, normalized size = 5.2 \begin{align*} \left [\frac{105 \,{\left (a^{3} e h -{\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (240 \, c^{3} f h x^{6} + 280 \,{\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \,{\left (7 \, c^{3} e g +{\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \,{\left (7 \, a c^{2} e h +{\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \,{\left (14 \, a c^{2} e g +{\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \,{\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \,{\left (a^{2} c e h +{\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt{c x^{2} + a}}{3360 \, c^{2}}, \frac{105 \,{\left (a^{3} e h -{\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (240 \, c^{3} f h x^{6} + 280 \,{\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \,{\left (7 \, c^{3} e g +{\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \,{\left (7 \, a c^{2} e h +{\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \,{\left (14 \, a c^{2} e g +{\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \,{\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \,{\left (a^{2} c e h +{\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt{c x^{2} + a}}{1680 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3360*(105*(a^3*e*h - (6*a^2*c*d - a^3*f)*g)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(24
0*c^3*f*h*x^6 + 280*(c^3*f*g + c^3*e*h)*x^5 + 336*a^2*c*e*g + 48*(7*c^3*e*g + (7*c^3*d + 8*a*c^2*f)*h)*x^4 + 7
0*(7*a*c^2*e*h + (6*c^3*d + 7*a*c^2*f)*g)*x^3 + 48*(14*a*c^2*e*g + (14*a*c^2*d + a^2*c*f)*h)*x^2 + 48*(7*a^2*c
*d - 2*a^3*f)*h + 105*(a^2*c*e*h + (10*a*c^2*d + a^2*c*f)*g)*x)*sqrt(c*x^2 + a))/c^2, 1/1680*(105*(a^3*e*h - (
6*a^2*c*d - a^3*f)*g)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (240*c^3*f*h*x^6 + 280*(c^3*f*g + c^3*e*h)
*x^5 + 336*a^2*c*e*g + 48*(7*c^3*e*g + (7*c^3*d + 8*a*c^2*f)*h)*x^4 + 70*(7*a*c^2*e*h + (6*c^3*d + 7*a*c^2*f)*
g)*x^3 + 48*(14*a*c^2*e*g + (14*a*c^2*d + a^2*c*f)*h)*x^2 + 48*(7*a^2*c*d - 2*a^3*f)*h + 105*(a^2*c*e*h + (10*
a*c^2*d + a^2*c*f)*g)*x)*sqrt(c*x^2 + a))/c^2]

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Sympy [A]  time = 25.5837, size = 768, normalized size = 3.61 \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+a)**(3/2)*(f*x**2+e*x+d),x)

[Out]

a**(5/2)*e*h*x/(16*c*sqrt(1 + c*x**2/a)) + a**(5/2)*f*g*x/(16*c*sqrt(1 + c*x**2/a)) + a**(3/2)*d*g*x*sqrt(1 +
c*x**2/a)/2 + a**(3/2)*d*g*x/(8*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*e*h*x**3/(48*sqrt(1 + c*x**2/a)) + 17*a**(3/
2)*f*g*x**3/(48*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*c*d*g*x**3/(8*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*e*h*x**5/(24*
sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*f*g*x**5/(24*sqrt(1 + c*x**2/a)) - a**3*e*h*asinh(sqrt(c)*x/sqrt(a))/(16*c*
*(3/2)) - a**3*f*g*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2)) + 3*a**2*d*g*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) +
a*d*h*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*e*g*Piecewise((sqrt(a)*x**2
/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*f*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*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*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*f*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**2*d*g*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*e*h*x**7/(6*sqrt(a)*sqrt(1 +
c*x**2/a)) + c**2*f*g*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.17146, size = 356, normalized size = 1.67 \begin{align*} \frac{1}{1680} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (6 \, c f h x + \frac{7 \,{\left (c^{6} f g + c^{6} h e\right )}}{c^{5}}\right )} x + \frac{6 \,{\left (7 \, c^{6} d h + 8 \, a c^{5} f h + 7 \, c^{6} g e\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (6 \, c^{6} d g + 7 \, a c^{5} f g + 7 \, a c^{5} h e\right )}}{c^{5}}\right )} x + \frac{24 \,{\left (14 \, a c^{5} d h + a^{2} c^{4} f h + 14 \, a c^{5} g e\right )}}{c^{5}}\right )} x + \frac{105 \,{\left (10 \, a c^{5} d g + a^{2} c^{4} f g + a^{2} c^{4} h e\right )}}{c^{5}}\right )} x + \frac{48 \,{\left (7 \, a^{2} c^{4} d h - 2 \, a^{3} c^{3} f h + 7 \, a^{2} c^{4} g e\right )}}{c^{5}}\right )} - \frac{{\left (6 \, a^{2} c d g - a^{3} f g - a^{3} h e\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*sqrt(c*x^2 + a)*((2*((4*(5*(6*c*f*h*x + 7*(c^6*f*g + c^6*h*e)/c^5)*x + 6*(7*c^6*d*h + 8*a*c^5*f*h + 7*c
^6*g*e)/c^5)*x + 35*(6*c^6*d*g + 7*a*c^5*f*g + 7*a*c^5*h*e)/c^5)*x + 24*(14*a*c^5*d*h + a^2*c^4*f*h + 14*a*c^5
*g*e)/c^5)*x + 105*(10*a*c^5*d*g + a^2*c^4*f*g + a^2*c^4*h*e)/c^5)*x + 48*(7*a^2*c^4*d*h - 2*a^3*c^3*f*h + 7*a
^2*c^4*g*e)/c^5) - 1/16*(6*a^2*c*d*g - a^3*f*g - a^3*h*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)