\(\int \frac {(d+e x)^{3/2} (a+b x+c x^2)}{\sqrt {f+g x}} \, dx\) [835]

Optimal result

Integrand size = 29, antiderivative size = 333 \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {(e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{64 e^{5/2} g^{9/2}} \]

[Out]

1/64*(-d*g+e*f)^2*(c*(3*d^2*g^2+10*d*e*f*g+35*e^2*f^2)+8*e*g*(6*a*e*g-b*(d*g+5*e*f)))*arctanh(g^(1/2)*(e*x+d)^
(1/2)/e^(1/2)/(g*x+f)^(1/2))/e^(5/2)/g^(9/2)+1/96*(c*(3*d^2*g^2+10*d*e*f*g+35*e^2*f^2)+8*e*g*(6*a*e*g-b*(d*g+5
*e*f)))*(e*x+d)^(3/2)*(g*x+f)^(1/2)/e^2/g^3-1/24*(-8*b*e*g+9*c*d*g+7*c*e*f)*(e*x+d)^(5/2)*(g*x+f)^(1/2)/e^2/g^
2+1/4*c*(e*x+d)^(7/2)*(g*x+f)^(1/2)/e^2/g-1/64*(-d*g+e*f)*(c*(3*d^2*g^2+10*d*e*f*g+35*e^2*f^2)+8*e*g*(6*a*e*g-
b*(d*g+5*e*f)))*(e*x+d)^(1/2)*(g*x+f)^(1/2)/e^2/g^4

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {965, 81, 52, 65, 223, 212} \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {(e f-d g)^2 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac {\sqrt {d+e x} \sqrt {f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac {(d+e x)^{3/2} \sqrt {f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}-\frac {(d+e x)^{5/2} \sqrt {f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g} \]

[In]

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

[Out]

-1/64*((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])/(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 52

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 65

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 81

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 965

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])

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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] (verified)

Time = 0.82 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {\sqrt {d+e x} \sqrt {f+g x} \left (c \left (-9 d^3 g^3+3 d^2 e g^2 (-5 f+2 g x)+d e^2 g \left (145 f^2-92 f g x+72 g^2 x^2\right )+e^3 \left (-105 f^3+70 f^2 g x-56 f g^2 x^2+48 g^3 x^3\right )\right )+8 e g \left (6 a e g (-3 e f+5 d g+2 e g x)+b \left (3 d^2 g^2+2 d e g (-11 f+7 g x)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )\right )}{192 e^2 g^4}+\frac {(e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{64 e^{5/2} g^{9/2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*Sqrt[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)))))/(192*e^2*g^4) + ((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[e]*Sqrt[f + g*
x])/(Sqrt[g]*Sqrt[d + e*x])])/(64*e^(5/2)*g^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1206\) vs. \(2(295)=590\).

Time = 0.48 (sec) , antiderivative size = 1207, normalized size of antiderivative = 3.62

method	result	size
default	\(\text {Expression too large to display}\)	\(1207\)

[In]

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

[Out]

1/384*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(-184*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d*e^2*f*g^2*x+144*c*d*e^2*g^3*x^
2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-112*c*e^3*f*g^2*x^2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+192*((g*x+f)*(e*
x+d))^(1/2)*(e*g)^(1/2)*a*e^3*g^3*x+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+48*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*b*d^2*e*g^3+240*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*
b*e^3*f^2*g-288*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*a*e^3*f*g^2+480*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*a*d*e^
2*g^3-180*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+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+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^2-288*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^3-72*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+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+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+
128*b*e^3*g^3*x^2*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)+96*c*e^3*g^3*x^3*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)-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+224*((g*x+f)*(e*x+d))^
(1/2)*(e*g)^(1/2)*b*d*e^2*g^3*x-160*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*b*e^3*f*g^2*x-30*((g*x+f)*(e*x+d))^(1/
2)*(e*g)^(1/2)*c*d^2*e*f*g^2-18*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d^3*g^3-352*((g*x+f)*(e*x+d))^(1/2)*(e*g
)^(1/2)*b*d*e^2*f*g^2+290*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*d*e^2*f^2*g-210*((g*x+f)*(e*x+d))^(1/2)*(e*g)^
(1/2)*c*e^3*f^3+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+
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+12*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(
1/2)*c*d^2*e*g^3*x+140*((g*x+f)*(e*x+d))^(1/2)*(e*g)^(1/2)*c*e^3*f^2*g*x)/g^4/e^2/((g*x+f)*(e*x+d))^(1/2)/(e*g
)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.62 (sec) , antiderivative size = 852, normalized size of antiderivative = 2.56 \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\left [\frac {3 \, {\left (35 \, c e^{4} f^{4} - 20 \, {\left (3 \, c d e^{3} + 2 \, b e^{4}\right )} f^{3} g + 6 \, {\left (3 \, c d^{2} e^{2} + 12 \, b d e^{3} + 8 \, a e^{4}\right )} f^{2} g^{2} + 4 \, {\left (c d^{3} e - 6 \, b d^{2} e^{2} - 24 \, a d e^{3}\right )} f g^{3} + {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} g^{4}\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) + 4 \, {\left (48 \, c e^{4} g^{4} x^{3} - 105 \, c e^{4} f^{3} g + 5 \, {\left (29 \, c d e^{3} + 24 \, b e^{4}\right )} f^{2} g^{2} - {\left (15 \, c d^{2} e^{2} + 176 \, b d e^{3} + 144 \, a e^{4}\right )} f g^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} g^{4} - 8 \, {\left (7 \, c e^{4} f g^{3} - {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} g^{4}\right )} x^{2} + 2 \, {\left (35 \, c e^{4} f^{2} g^{2} - 2 \, {\left (23 \, c d e^{3} + 20 \, b e^{4}\right )} f g^{3} + {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} g^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{768 \, e^{3} g^{5}}, -\frac {3 \, {\left (35 \, c e^{4} f^{4} - 20 \, {\left (3 \, c d e^{3} + 2 \, b e^{4}\right )} f^{3} g + 6 \, {\left (3 \, c d^{2} e^{2} + 12 \, b d e^{3} + 8 \, a e^{4}\right )} f^{2} g^{2} + 4 \, {\left (c d^{3} e - 6 \, b d^{2} e^{2} - 24 \, a d e^{3}\right )} f g^{3} + {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} g^{4}\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c e^{4} g^{4} x^{3} - 105 \, c e^{4} f^{3} g + 5 \, {\left (29 \, c d e^{3} + 24 \, b e^{4}\right )} f^{2} g^{2} - {\left (15 \, c d^{2} e^{2} + 176 \, b d e^{3} + 144 \, a e^{4}\right )} f g^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} g^{4} - 8 \, {\left (7 \, c e^{4} f g^{3} - {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} g^{4}\right )} x^{2} + 2 \, {\left (35 \, c e^{4} f^{2} g^{2} - 2 \, {\left (23 \, c d e^{3} + 20 \, b e^{4}\right )} f g^{3} + {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} g^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{384 \, e^{3} g^{5}}\right ] \]

[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)]

Sympy [F]

\[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )}{\sqrt {f + g x}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d*g-e*f>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.44 \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {{\left (\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left (2 \, {\left (e x + d\right )} {\left (4 \, {\left (e x + d\right )} {\left (\frac {6 \, {\left (e x + d\right )} c}{e^{3} g} - \frac {7 \, c e^{7} f g^{5} + 9 \, c d e^{6} g^{6} - 8 \, b e^{7} g^{6}}{e^{9} g^{7}}\right )} + \frac {35 \, c e^{8} f^{2} g^{4} + 10 \, c d e^{7} f g^{5} - 40 \, b e^{8} f g^{5} + 3 \, c d^{2} e^{6} g^{6} - 8 \, b d e^{7} g^{6} + 48 \, a e^{8} g^{6}}{e^{9} g^{7}}\right )} - \frac {3 \, {\left (35 \, c e^{9} f^{3} g^{3} - 25 \, c d e^{8} f^{2} g^{4} - 40 \, b e^{9} f^{2} g^{4} - 7 \, c d^{2} e^{7} f g^{5} + 32 \, b d e^{8} f g^{5} + 48 \, a e^{9} f g^{5} - 3 \, c d^{3} e^{6} g^{6} + 8 \, b d^{2} e^{7} g^{6} - 48 \, a d e^{8} g^{6}\right )}}{e^{9} g^{7}}\right )} \sqrt {e x + d} - \frac {3 \, {\left (35 \, c e^{4} f^{4} - 60 \, c d e^{3} f^{3} g - 40 \, b e^{4} f^{3} g + 18 \, c d^{2} e^{2} f^{2} g^{2} + 72 \, b d e^{3} f^{2} g^{2} + 48 \, a e^{4} f^{2} g^{2} + 4 \, c d^{3} e f g^{3} - 24 \, b d^{2} e^{2} f g^{3} - 96 \, a d e^{3} f g^{3} + 3 \, c d^{4} g^{4} - 8 \, b d^{3} e g^{4} + 48 \, a d^{2} e^{2} g^{4}\right )} \log \left ({\left | -\sqrt {e g} \sqrt {e x + d} + \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \right |}\right )}{\sqrt {e g} e^{2} g^{4}}\right )} e}{192 \, {\left | e \right |}} \]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )}{\sqrt {f+g\,x}} \,d x \]

[In]

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

[Out]

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