3.835 \(\int \frac{(d+e x)^{3/2} (a+b x+c x^2)}{\sqrt{f+g x}} \, dx\)

Optimal. Leaf size=333 \[ \frac{(d+e x)^{3/2} \sqrt{f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}-\frac{\sqrt{d+e x} \sqrt{f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac{(d+e x)^{5/2} \sqrt{f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g} \]

[Out]

-((e*f - d*g)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Sqrt[d + e*x]*Sqrt
[f + g*x])/(64*e^2*g^4) + ((c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*(d +
e*x)^(3/2)*Sqrt[f + g*x])/(96*e^2*g^3) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[f + g*x])/(24*e^2
*g^2) + (c*(d + e*x)^(7/2)*Sqrt[f + g*x])/(4*e^2*g) + ((e*f - d*g)^2*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2)
+ 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(64*e^(5/2)*g^(
9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.353488, antiderivative size = 333, 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 = {951, 80, 50, 63, 217, 206} \[ \frac{(d+e x)^{3/2} \sqrt{f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}-\frac{\sqrt{d+e x} \sqrt{f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac{(d+e x)^{5/2} \sqrt{f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*f - d*g)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Sqrt[d + e*x]*Sqrt
[f + g*x])/(64*e^2*g^4) + ((c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*(d +
e*x)^(3/2)*Sqrt[f + g*x])/(96*e^2*g^3) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[f + g*x])/(24*e^2
*g^2) + (c*(d + e*x)^(7/2)*Sqrt[f + g*x])/(4*e^2*g) + ((e*f - d*g)^2*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2)
+ 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(64*e^(5/2)*g^(
9/2))

Rule 951

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x
] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 \frac{(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt{f+g x}} \, dx &=\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}+\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} \left (8 a e^2 g-c d (7 e f+d g)\right )-\frac{1}{2} e (7 c e f+9 c d g-8 b e g) x\right )}{\sqrt{f+g x}} \, dx}{4 e^2 g}\\ &=-\frac{(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{f+g x}}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}+\frac{\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \int \frac{(d+e x)^{3/2}}{\sqrt{f+g x}} \, dx}{48 e^2 g^2}\\ &=\frac{\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt{f+g x}}{96 e^2 g^3}-\frac{(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{f+g x}}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}-\frac{\left ((e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{f+g x}} \, dx}{64 e^2 g^3}\\ &=-\frac{(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{64 e^2 g^4}+\frac{\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt{f+g x}}{96 e^2 g^3}-\frac{(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{f+g x}}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}+\frac{\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{128 e^2 g^4}\\ &=-\frac{(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{64 e^2 g^4}+\frac{\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt{f+g x}}{96 e^2 g^3}-\frac{(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{f+g x}}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}+\frac{\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{d g}{e}+\frac{g x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{64 e^3 g^4}\\ &=-\frac{(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{64 e^2 g^4}+\frac{\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt{f+g x}}{96 e^2 g^3}-\frac{(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{f+g x}}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}+\frac{\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{e}} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{64 e^3 g^4}\\ &=-\frac{(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt{d+e x} \sqrt{f+g x}}{64 e^2 g^4}+\frac{\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt{f+g x}}{96 e^2 g^3}-\frac{(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{f+g x}}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}+\frac{(e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{64 e^{5/2} g^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.67101, size = 313, normalized size = 0.94 \[ \frac{3 (e f-d g)^{5/2} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )-e \sqrt{g} \sqrt{d+e x} (f+g x) \left (c \left (3 d^2 e g^2 (5 f-2 g x)+9 d^3 g^3+d e^2 g \left (-145 f^2+92 f g x-72 g^2 x^2\right )+e^3 \left (-70 f^2 g x+105 f^3+56 f g^2 x^2-48 g^3 x^3\right )\right )-8 e g \left (6 a e g (5 d g-3 e f+2 e g x)+b \left (3 d^2 g^2+2 d e g (7 g x-11 f)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )\right )}{192 e^3 g^{9/2} \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(c*(9*d^3*g^3 + 3*d^2*e*g^2*(5*f - 2*g*x) + d*e^2*g*(-145*f^2 + 92*f*g*x
- 72*g^2*x^2) + e^3*(105*f^3 - 70*f^2*g*x + 56*f*g^2*x^2 - 48*g^3*x^3)) - 8*e*g*(6*a*e*g*(-3*e*f + 5*d*g + 2*e
*g*x) + b*(3*d^2*g^2 + 2*d*e*g*(-11*f + 7*g*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2))))) + 3*(e*f - d*g)^(5/2)
*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]
*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(192*e^3*g^(9/2)*Sqrt[f + g*x])

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Maple [B]  time = 0.286, size = 1207, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(c*x^2+b*x+a)/(g*x+f)^(1/2),x)

[Out]

1/384*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(144*x^2*c*d*e^2*g^3*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-112*x^2*c*e^3*f*g^2
*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-352*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*b*d*e^2*f*g^2+290*(e*g)^(1/2)*((g
*x+f)*(e*x+d))^(1/2)*c*d*e^2*f^2*g+224*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*x*b*d*e^2*g^3-160*(e*g)^(1/2)*((g*x
+f)*(e*x+d))^(1/2)*x*b*e^3*f*g^2+12*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*x*c*d^2*e*g^3+140*(e*g)^(1/2)*((g*x+f)
*(e*x+d))^(1/2)*x*c*e^3*f^2*g-30*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c*d^2*e*f*g^2+144*ln(1/2*(2*e*g*x+2*((g*x
+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*d^2*e^2*g^4+105*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/
2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*e^4*f^4+9*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f
)/(e*g)^(1/2))*c*d^4*g^4+144*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*e^4
*f^2*g^2-120*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*e^4*f^3*g-210*(e*g)
^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c*e^3*f^3-24*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*
g)^(1/2))*b*d^3*e*g^4-18*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c*d^3*g^3-184*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)
*x*c*d*e^2*f*g^2-288*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*a*d*e^3*f*g^3
+240*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*b*e^3*f^2*g+96*x^3*c*e^3*g^3*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+128*
x^2*b*e^3*g^3*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+192*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*x*a*e^3*g^3+48*(e*g)
^(1/2)*((g*x+f)*(e*x+d))^(1/2)*b*d^2*e*g^3+12*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(
e*g)^(1/2))*c*d^3*e*f*g^3-72*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*d^2
*e^2*f*g^3+216*ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*b*d*e^3*f^2*g^2+54*
ln(1/2*(2*e*g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f^2*g^2-180*ln(1/2*(2*e*
g*x+2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^3*f^3*g+480*(e*g)^(1/2)*((g*x+f)*(e*x+d)
)^(1/2)*a*d*e^2*g^3-288*(e*g)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*a*e^3*f*g^2)/e^2/g^4/((g*x+f)*(e*x+d))^(1/2)/(e*g)
^(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((e*x+d)^(3/2)*(c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.15194, size = 1912, normalized size = 5.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(3*(35*c*e^4*f^4 - 20*(3*c*d*e^3 + 2*b*e^4)*f^3*g + 6*(3*c*d^2*e^2 + 12*b*d*e^3 + 8*a*e^4)*f^2*g^2 + 4*
(c*d^3*e - 6*b*d^2*e^2 - 24*a*d*e^3)*f*g^3 + (3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*g^4)*sqrt(e*g)*log(8*e^2*g^2
*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 4*(2*e*g*x + e*f + d*g)*sqrt(e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(e^2*
f*g + d*e*g^2)*x) + 4*(48*c*e^4*g^4*x^3 - 105*c*e^4*f^3*g + 5*(29*c*d*e^3 + 24*b*e^4)*f^2*g^2 - (15*c*d^2*e^2
+ 176*b*d*e^3 + 144*a*e^4)*f*g^3 - 3*(3*c*d^3*e - 8*b*d^2*e^2 - 80*a*d*e^3)*g^4 - 8*(7*c*e^4*f*g^3 - (9*c*d*e^
3 + 8*b*e^4)*g^4)*x^2 + 2*(35*c*e^4*f^2*g^2 - 2*(23*c*d*e^3 + 20*b*e^4)*f*g^3 + (3*c*d^2*e^2 + 56*b*d*e^3 + 48
*a*e^4)*g^4)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^5), -1/384*(3*(35*c*e^4*f^4 - 20*(3*c*d*e^3 + 2*b*e^4)*f^3
*g + 6*(3*c*d^2*e^2 + 12*b*d*e^3 + 8*a*e^4)*f^2*g^2 + 4*(c*d^3*e - 6*b*d^2*e^2 - 24*a*d*e^3)*f*g^3 + (3*c*d^4
- 8*b*d^3*e + 48*a*d^2*e^2)*g^4)*sqrt(-e*g)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt(-e*g)*sqrt(e*x + d)*sqrt(g*x
+ f)/(e^2*g^2*x^2 + d*e*f*g + (e^2*f*g + d*e*g^2)*x)) - 2*(48*c*e^4*g^4*x^3 - 105*c*e^4*f^3*g + 5*(29*c*d*e^3
+ 24*b*e^4)*f^2*g^2 - (15*c*d^2*e^2 + 176*b*d*e^3 + 144*a*e^4)*f*g^3 - 3*(3*c*d^3*e - 8*b*d^2*e^2 - 80*a*d*e^
3)*g^4 - 8*(7*c*e^4*f*g^3 - (9*c*d*e^3 + 8*b*e^4)*g^4)*x^2 + 2*(35*c*e^4*f^2*g^2 - 2*(23*c*d*e^3 + 20*b*e^4)*f
*g^3 + (3*c*d^2*e^2 + 56*b*d*e^3 + 48*a*e^4)*g^4)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.28985, size = 605, normalized size = 1.82 \begin{align*} \frac{1}{192} \, \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}}{\left (2 \,{\left (4 \,{\left (x e + d\right )}{\left (\frac{6 \,{\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac{{\left (9 \, c d g^{6} e^{6} + 7 \, c f g^{5} e^{7} - 8 \, b g^{6} e^{7}\right )} e^{\left (-9\right )}}{g^{7}}\right )} + \frac{{\left (3 \, c d^{2} g^{6} e^{6} + 10 \, c d f g^{5} e^{7} - 8 \, b d g^{6} e^{7} + 35 \, c f^{2} g^{4} e^{8} - 40 \, b f g^{5} e^{8} + 48 \, a g^{6} e^{8}\right )} e^{\left (-9\right )}}{g^{7}}\right )}{\left (x e + d\right )} + \frac{3 \,{\left (3 \, c d^{3} g^{6} e^{6} + 7 \, c d^{2} f g^{5} e^{7} - 8 \, b d^{2} g^{6} e^{7} + 25 \, c d f^{2} g^{4} e^{8} - 32 \, b d f g^{5} e^{8} + 48 \, a d g^{6} e^{8} - 35 \, c f^{3} g^{3} e^{9} + 40 \, b f^{2} g^{4} e^{9} - 48 \, a f g^{5} e^{9}\right )} e^{\left (-9\right )}}{g^{7}}\right )} \sqrt{x e + d} - \frac{{\left (3 \, c d^{4} g^{4} + 4 \, c d^{3} f g^{3} e - 8 \, b d^{3} g^{4} e + 18 \, c d^{2} f^{2} g^{2} e^{2} - 24 \, b d^{2} f g^{3} e^{2} + 48 \, a d^{2} g^{4} e^{2} - 60 \, c d f^{3} g e^{3} + 72 \, b d f^{2} g^{2} e^{3} - 96 \, a d f g^{3} e^{3} + 35 \, c f^{4} e^{4} - 40 \, b f^{3} g e^{4} + 48 \, a f^{2} g^{2} e^{4}\right )} e^{\left (-\frac{5}{2}\right )} \log \left ({\left | -\sqrt{x e + d} \sqrt{g} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{64 \, g^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*sqrt((x*e + d)*g*e - d*g*e + f*e^2)*(2*(4*(x*e + d)*(6*(x*e + d)*c*e^(-3)/g - (9*c*d*g^6*e^6 + 7*c*f*g^5
*e^7 - 8*b*g^6*e^7)*e^(-9)/g^7) + (3*c*d^2*g^6*e^6 + 10*c*d*f*g^5*e^7 - 8*b*d*g^6*e^7 + 35*c*f^2*g^4*e^8 - 40*
b*f*g^5*e^8 + 48*a*g^6*e^8)*e^(-9)/g^7)*(x*e + d) + 3*(3*c*d^3*g^6*e^6 + 7*c*d^2*f*g^5*e^7 - 8*b*d^2*g^6*e^7 +
25*c*d*f^2*g^4*e^8 - 32*b*d*f*g^5*e^8 + 48*a*d*g^6*e^8 - 35*c*f^3*g^3*e^9 + 40*b*f^2*g^4*e^9 - 48*a*f*g^5*e^9
)*e^(-9)/g^7)*sqrt(x*e + d) - 1/64*(3*c*d^4*g^4 + 4*c*d^3*f*g^3*e - 8*b*d^3*g^4*e + 18*c*d^2*f^2*g^2*e^2 - 24*
b*d^2*f*g^3*e^2 + 48*a*d^2*g^4*e^2 - 60*c*d*f^3*g*e^3 + 72*b*d*f^2*g^2*e^3 - 96*a*d*f*g^3*e^3 + 35*c*f^4*e^4 -
40*b*f^3*g*e^4 + 48*a*f^2*g^2*e^4)*e^(-5/2)*log(abs(-sqrt(x*e + d)*sqrt(g)*e^(1/2) + sqrt((x*e + d)*g*e - d*g
*e + f*e^2)))/g^(9/2)