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

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

[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

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Rubi [A]
time = 0.22, antiderivative size = 333, normalized size of antiderivative = 1.00, 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 = {965, 81, 52, 65, 223, 212} \begin {gather*} -\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 {(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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 ) \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*}

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Mathematica [A]
time = 0.82, size = 289, normalized size = 0.87 \begin {gather*} \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 ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{64 e^{5/2} g^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1206\) vs. \(2(295)=590\).
time = 0.07, size = 1207, normalized size = 3.62

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

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,method=_RETURNVERBOSE)

[Out]

1/384*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(-112*c*e^3*f*g^2*x^2*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)-184*(e*g)^(1/2)*((
e*x+d)*(g*x+f))^(1/2)*c*d*e^2*f*g^2*x-30*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*c*d^2*e*f*g^2+192*(e*g)^(1/2)*((e
*x+d)*(g*x+f))^(1/2)*a*e^3*g^3*x+48*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*b*d^2*e*g^3+240*(e*g)^(1/2)*((e*x+d)*(
g*x+f))^(1/2)*b*e^3*f^2*g-180*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(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)*((e*x+d)*(g*x+f))^(1/2)*a*d*e^2*g^3-288*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*a*e^3*f*
g^2-288*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(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*((e*x+d)*(g*x+f))^(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*((e*x
+d)*(g*x+f))^(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*((e*x+d)*(g*x+f))^(1
/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f^2*g^2+9*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(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*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/
2))*c*e^4*f^4+96*c*e^3*g^3*x^3*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)+128*b*e^3*g^3*x^2*(e*g)^(1/2)*((e*x+d)*(g*x
+f))^(1/2)-352*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*b*d*e^2*f*g^2+290*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*c*d*e
^2*f^2*g+144*c*d*e^2*g^3*x^2*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)+144*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(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*((e*x+d)*(g*x+f))^(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*((e*x+d)*(g*x+f))^(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)*((e*x+d)*(g*x+f))^(1/2)*c*e^3*f^3+12*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*
(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*e*f*g^3+224*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*b*d*e^2*g^3*x-160*(e*g
)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*b*e^3*f*g^2*x+12*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*c*d^2*e*g^3*x+140*(e*g)^(
1/2)*((e*x+d)*(g*x+f))^(1/2)*c*e^3*f^2*g*x-24*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(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)*((e*x+d)*(g*x+f))^(1/2)*c*d^3*g^3)/g^4/e^2/((e*x+d)*(g*x+f))^(1/2)/(e*g
)^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> 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?` fo
r more detai

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Fricas [A]
time = 5.56, size = 844, normalized size = 2.53 \begin {gather*} \left [\frac {{\left (3 \, {\left (3 \, c d^{4} g^{4} + {\left (35 \, c f^{4} - 40 \, b f^{3} g + 48 \, a f^{2} g^{2}\right )} e^{4} - 12 \, {\left (5 \, c d f^{3} g - 6 \, b d f^{2} g^{2} + 8 \, a d f g^{3}\right )} e^{3} + 6 \, {\left (3 \, c d^{2} f^{2} g^{2} - 4 \, b d^{2} f g^{3} + 8 \, a d^{2} g^{4}\right )} e^{2} + 4 \, {\left (c d^{3} f g^{3} - 2 \, b d^{3} g^{4}\right )} e\right )} \sqrt {g} e^{\frac {1}{2}} \log \left (d^{2} g^{2} + 4 \, {\left (d g + {\left (2 \, g x + f\right )} e\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} + {\left (8 \, g^{2} x^{2} + 8 \, f g x + f^{2}\right )} e^{2} + 2 \, {\left (4 \, d g^{2} x + 3 \, d f g\right )} e\right ) - 4 \, {\left (9 \, c d^{3} g^{4} e - {\left (48 \, c g^{4} x^{3} - 105 \, c f^{3} g + 120 \, b f^{2} g^{2} - 144 \, a f g^{3} - 8 \, {\left (7 \, c f g^{3} - 8 \, b g^{4}\right )} x^{2} + 2 \, {\left (35 \, c f^{2} g^{2} - 40 \, b f g^{3} + 48 \, a g^{4}\right )} x\right )} e^{4} - {\left (72 \, c d g^{4} x^{2} + 145 \, c d f^{2} g^{2} - 176 \, b d f g^{3} + 240 \, a d g^{4} - 4 \, {\left (23 \, c d f g^{3} - 28 \, b d g^{4}\right )} x\right )} e^{3} - 3 \, {\left (2 \, c d^{2} g^{4} x - 5 \, c d^{2} f g^{3} + 8 \, b d^{2} g^{4}\right )} e^{2}\right )} \sqrt {g x + f} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{768 \, g^{5}}, -\frac {{\left (3 \, {\left (3 \, c d^{4} g^{4} + {\left (35 \, c f^{4} - 40 \, b f^{3} g + 48 \, a f^{2} g^{2}\right )} e^{4} - 12 \, {\left (5 \, c d f^{3} g - 6 \, b d f^{2} g^{2} + 8 \, a d f g^{3}\right )} e^{3} + 6 \, {\left (3 \, c d^{2} f^{2} g^{2} - 4 \, b d^{2} f g^{3} + 8 \, a d^{2} g^{4}\right )} e^{2} + 4 \, {\left (c d^{3} f g^{3} - 2 \, b d^{3} g^{4}\right )} e\right )} \sqrt {-g e} \arctan \left (\frac {{\left (d g + {\left (2 \, g x + f\right )} e\right )} \sqrt {g x + f} \sqrt {-g e} \sqrt {x e + d}}{2 \, {\left ({\left (g^{2} x^{2} + f g x\right )} e^{2} + {\left (d g^{2} x + d f g\right )} e\right )}}\right ) + 2 \, {\left (9 \, c d^{3} g^{4} e - {\left (48 \, c g^{4} x^{3} - 105 \, c f^{3} g + 120 \, b f^{2} g^{2} - 144 \, a f g^{3} - 8 \, {\left (7 \, c f g^{3} - 8 \, b g^{4}\right )} x^{2} + 2 \, {\left (35 \, c f^{2} g^{2} - 40 \, b f g^{3} + 48 \, a g^{4}\right )} x\right )} e^{4} - {\left (72 \, c d g^{4} x^{2} + 145 \, c d f^{2} g^{2} - 176 \, b d f g^{3} + 240 \, a d g^{4} - 4 \, {\left (23 \, c d f g^{3} - 28 \, b d g^{4}\right )} x\right )} e^{3} - 3 \, {\left (2 \, c d^{2} g^{4} x - 5 \, c d^{2} f g^{3} + 8 \, b d^{2} g^{4}\right )} e^{2}\right )} \sqrt {g x + f} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{384 \, g^{5}}\right ] \end {gather*}

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*(3*c*d^4*g^4 + (35*c*f^4 - 40*b*f^3*g + 48*a*f^2*g^2)*e^4 - 12*(5*c*d*f^3*g - 6*b*d*f^2*g^2 + 8*a*d*
f*g^3)*e^3 + 6*(3*c*d^2*f^2*g^2 - 4*b*d^2*f*g^3 + 8*a*d^2*g^4)*e^2 + 4*(c*d^3*f*g^3 - 2*b*d^3*g^4)*e)*sqrt(g)*
e^(1/2)*log(d^2*g^2 + 4*(d*g + (2*g*x + f)*e)*sqrt(g*x + f)*sqrt(x*e + d)*sqrt(g)*e^(1/2) + (8*g^2*x^2 + 8*f*g
*x + f^2)*e^2 + 2*(4*d*g^2*x + 3*d*f*g)*e) - 4*(9*c*d^3*g^4*e - (48*c*g^4*x^3 - 105*c*f^3*g + 120*b*f^2*g^2 -
144*a*f*g^3 - 8*(7*c*f*g^3 - 8*b*g^4)*x^2 + 2*(35*c*f^2*g^2 - 40*b*f*g^3 + 48*a*g^4)*x)*e^4 - (72*c*d*g^4*x^2
+ 145*c*d*f^2*g^2 - 176*b*d*f*g^3 + 240*a*d*g^4 - 4*(23*c*d*f*g^3 - 28*b*d*g^4)*x)*e^3 - 3*(2*c*d^2*g^4*x - 5*
c*d^2*f*g^3 + 8*b*d^2*g^4)*e^2)*sqrt(g*x + f)*sqrt(x*e + d))*e^(-3)/g^5, -1/384*(3*(3*c*d^4*g^4 + (35*c*f^4 -
40*b*f^3*g + 48*a*f^2*g^2)*e^4 - 12*(5*c*d*f^3*g - 6*b*d*f^2*g^2 + 8*a*d*f*g^3)*e^3 + 6*(3*c*d^2*f^2*g^2 - 4*b
*d^2*f*g^3 + 8*a*d^2*g^4)*e^2 + 4*(c*d^3*f*g^3 - 2*b*d^3*g^4)*e)*sqrt(-g*e)*arctan(1/2*(d*g + (2*g*x + f)*e)*s
qrt(g*x + f)*sqrt(-g*e)*sqrt(x*e + d)/((g^2*x^2 + f*g*x)*e^2 + (d*g^2*x + d*f*g)*e)) + 2*(9*c*d^3*g^4*e - (48*
c*g^4*x^3 - 105*c*f^3*g + 120*b*f^2*g^2 - 144*a*f*g^3 - 8*(7*c*f*g^3 - 8*b*g^4)*x^2 + 2*(35*c*f^2*g^2 - 40*b*f
*g^3 + 48*a*g^4)*x)*e^4 - (72*c*d*g^4*x^2 + 145*c*d*f^2*g^2 - 176*b*d*f*g^3 + 240*a*d*g^4 - 4*(23*c*d*f*g^3 -
28*b*d*g^4)*x)*e^3 - 3*(2*c*d^2*g^4*x - 5*c*d^2*f*g^3 + 8*b*d^2*g^4)*e^2)*sqrt(g*x + f)*sqrt(x*e + d))*e^(-3)/
g^5]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1048 vs. \(2 (307) = 614\).
time = 4.67, size = 1048, normalized size = 3.15 \begin {gather*} -\frac {\frac {192 \, {\left (\frac {{\left (d g^{2} - f g e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{\sqrt {g}} - \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \sqrt {g x + f}\right )} a d {\left | g \right |}}{g^{2}} - \frac {8 \, {\left (\sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \sqrt {g x + f} {\left (2 \, {\left (g x + f\right )} {\left (\frac {4 \, {\left (g x + f\right )}}{g^{2}} + \frac {{\left (d g^{6} e^{3} - 13 \, f g^{5} e^{4}\right )} e^{\left (-4\right )}}{g^{7}}\right )} - \frac {3 \, {\left (d^{2} g^{7} e^{2} + 2 \, d f g^{6} e^{3} - 11 \, f^{2} g^{5} e^{4}\right )} e^{\left (-4\right )}}{g^{7}}\right )} - \frac {3 \, {\left (d^{3} g^{3} + d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} - 5 \, f^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{g^{\frac {3}{2}}}\right )} c d {\left | g \right |}}{g^{2}} - \frac {8 \, {\left (\sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \sqrt {g x + f} {\left (2 \, {\left (g x + f\right )} {\left (\frac {4 \, {\left (g x + f\right )}}{g^{2}} + \frac {{\left (d g^{6} e^{3} - 13 \, f g^{5} e^{4}\right )} e^{\left (-4\right )}}{g^{7}}\right )} - \frac {3 \, {\left (d^{2} g^{7} e^{2} + 2 \, d f g^{6} e^{3} - 11 \, f^{2} g^{5} e^{4}\right )} e^{\left (-4\right )}}{g^{7}}\right )} - \frac {3 \, {\left (d^{3} g^{3} + d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} - 5 \, f^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{g^{\frac {3}{2}}}\right )} b {\left | g \right |} e}{g^{2}} - \frac {{\left (\sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} {\left (2 \, {\left (g x + f\right )} {\left (4 \, {\left (g x + f\right )} {\left (\frac {6 \, {\left (g x + f\right )}}{g^{3}} + \frac {{\left (d g^{12} e^{5} - 25 \, f g^{11} e^{6}\right )} e^{\left (-6\right )}}{g^{14}}\right )} - \frac {{\left (5 \, d^{2} g^{13} e^{4} + 14 \, d f g^{12} e^{5} - 163 \, f^{2} g^{11} e^{6}\right )} e^{\left (-6\right )}}{g^{14}}\right )} + \frac {3 \, {\left (5 \, d^{3} g^{14} e^{3} + 9 \, d^{2} f g^{13} e^{4} + 15 \, d f^{2} g^{12} e^{5} - 93 \, f^{3} g^{11} e^{6}\right )} e^{\left (-6\right )}}{g^{14}}\right )} \sqrt {g x + f} + \frac {3 \, {\left (5 \, d^{4} g^{4} + 4 \, d^{3} f g^{3} e + 6 \, d^{2} f^{2} g^{2} e^{2} + 20 \, d f^{3} g e^{3} - 35 \, f^{4} e^{4}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{g^{\frac {5}{2}}}\right )} c {\left | g \right |} e}{g^{2}} - \frac {48 \, {\left (\frac {{\left (d^{2} g^{3} + 2 \, d f g^{2} e - 3 \, f^{2} g e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{\sqrt {g}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} {\left (2 \, g x + {\left (d g e - 5 \, f e^{2}\right )} e^{\left (-2\right )} + 2 \, f\right )} \sqrt {g x + f}\right )} b d {\left | g \right |}}{g^{3}} - \frac {48 \, {\left (\frac {{\left (d^{2} g^{3} + 2 \, d f g^{2} e - 3 \, f^{2} g e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{\sqrt {g}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} {\left (2 \, g x + {\left (d g e - 5 \, f e^{2}\right )} e^{\left (-2\right )} + 2 \, f\right )} \sqrt {g x + f}\right )} a {\left | g \right |} e}{g^{3}}}{192 \, g} \end {gather*}

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*(192*((d*g^2 - f*g*e)*e^(-1/2)*log(abs(-sqrt(g*x + f)*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*x + f)*g*e - f*
g*e)))/sqrt(g) - sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)*sqrt(g*x + f))*a*d*abs(g)/g^2 - 8*(sqrt(d*g^2 + (g*x + f)
*g*e - f*g*e)*sqrt(g*x + f)*(2*(g*x + f)*(4*(g*x + f)/g^2 + (d*g^6*e^3 - 13*f*g^5*e^4)*e^(-4)/g^7) - 3*(d^2*g^
7*e^2 + 2*d*f*g^6*e^3 - 11*f^2*g^5*e^4)*e^(-4)/g^7) - 3*(d^3*g^3 + d^2*f*g^2*e + 3*d*f^2*g*e^2 - 5*f^3*e^3)*e^
(-5/2)*log(abs(-sqrt(g*x + f)*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)))/g^(3/2))*c*d*abs(g)/g^2
- 8*(sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)*sqrt(g*x + f)*(2*(g*x + f)*(4*(g*x + f)/g^2 + (d*g^6*e^3 - 13*f*g^5*e
^4)*e^(-4)/g^7) - 3*(d^2*g^7*e^2 + 2*d*f*g^6*e^3 - 11*f^2*g^5*e^4)*e^(-4)/g^7) - 3*(d^3*g^3 + d^2*f*g^2*e + 3*
d*f^2*g*e^2 - 5*f^3*e^3)*e^(-5/2)*log(abs(-sqrt(g*x + f)*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)
))/g^(3/2))*b*abs(g)*e/g^2 - (sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)*(2*(g*x + f)*(4*(g*x + f)*(6*(g*x + f)/g^3 +
(d*g^12*e^5 - 25*f*g^11*e^6)*e^(-6)/g^14) - (5*d^2*g^13*e^4 + 14*d*f*g^12*e^5 - 163*f^2*g^11*e^6)*e^(-6)/g^14
) + 3*(5*d^3*g^14*e^3 + 9*d^2*f*g^13*e^4 + 15*d*f^2*g^12*e^5 - 93*f^3*g^11*e^6)*e^(-6)/g^14)*sqrt(g*x + f) + 3
*(5*d^4*g^4 + 4*d^3*f*g^3*e + 6*d^2*f^2*g^2*e^2 + 20*d*f^3*g*e^3 - 35*f^4*e^4)*e^(-7/2)*log(abs(-sqrt(g*x + f)
*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)))/g^(5/2))*c*abs(g)*e/g^2 - 48*((d^2*g^3 + 2*d*f*g^2*e
- 3*f^2*g*e^2)*e^(-3/2)*log(abs(-sqrt(g*x + f)*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)))/sqrt(g)
+ sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)*(2*g*x + (d*g*e - 5*f*e^2)*e^(-2) + 2*f)*sqrt(g*x + f))*b*d*abs(g)/g^3
- 48*((d^2*g^3 + 2*d*f*g^2*e - 3*f^2*g*e^2)*e^(-3/2)*log(abs(-sqrt(g*x + f)*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*
x + f)*g*e - f*g*e)))/sqrt(g) + sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)*(2*g*x + (d*g*e - 5*f*e^2)*e^(-2) + 2*f)*s
qrt(g*x + f))*a*abs(g)*e/g^3)/g

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )}{\sqrt {f+g\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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