3.25.72 \(\int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx\) [2472]

Optimal. Leaf size=444 \[ \frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\left (5 A e (2 c d-b e)-B \left (4 c d^2+e (b d-6 a e)\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

[Out]

-1/16*(b^3*e^2*(5*A*e+B*d)-2*b^2*e*(9*A*c*d*e+3*B*a*e^2+2*B*c*d^2)+4*b*c*(-3*A*a*e^3+6*A*c*d^2*e+5*B*a*d*e^2+2
*B*c*d^3)-8*c*(A*c*d*(-3*a*e^2+2*c*d^2)+a*B*e*(-a*e^2+4*c*d^2)))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2
-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(7/2)+1/3*(-A*e+B*d)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b
*d*e+c*d^2)/(e*x+d)^3-1/12*(5*A*e*(-b*e+2*c*d)-B*(4*c*d^2+e*(-6*a*e+b*d)))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*
d^2)^2/(e*x+d)^2+1/24*(B*(8*c^2*d^3+2*c*d*e*(-26*a*e+5*b*d)-3*b*e^2*(-6*a*e+b*d))-A*e*(44*c^2*d^2+15*b^2*e^2-4
*c*e*(4*a*e+11*b*d)))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)

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Rubi [A]
time = 0.53, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {848, 820, 738, 212} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((4*B*c*d^2 + B*e*(b*d - 6*a*e)
- 5*A*e*(2*c*d - b*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*
c*d*e*(5*b*d - 26*a*e) - 3*b*e^2*(b*d - 6*a*e)) - A*e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e)))*Sqrt
[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - ((b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9
*A*c*d*e + 3*a*B*e^2) + 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) - 8*c*(A*c*d*(2*c*d^2 - 3*a*
e^2) + a*B*e*(4*c*d^2 - a*e^2)))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a
+ b*x + c*x^2])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\int \frac {\frac {1}{2} (b B d-6 A c d+5 A b e-6 a B e)-2 c (B d-A e) x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\int \frac {\frac {1}{4} \left (3 b^2 e (B d+5 A e)+8 c \left (3 A c d^2+5 a B d e-2 a A e^2\right )-2 b \left (4 B c d^2+17 A c d e+9 a B e^2\right )\right )+\frac {1}{2} c \left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{6 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 10.64, size = 434, normalized size = 0.98 \begin {gather*} \frac {\frac {2 (B d-A e) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)+5 A e (-2 c d+b e)\right ) \sqrt {a+x (b+c x)}}{2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )+A e \left (-44 c^2 d^2-15 b^2 e^2+4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {3 \left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )+8 c \left (a B e \left (-4 c d^2+a e^2\right )+A c d \left (-2 c d^2+3 a e^2\right )\right )\right ) \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}}{6 \left (c d^2+e (-b d+a e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

((2*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + ((4*B*c*d^2 + B*e*(b*d - 6*a*e
) + 5*A*e*(-2*c*d + b*e))*Sqrt[a + x*(b + c*x)])/(2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*c*d*e*(5*b*d - 26*a*e) -
3*b*e^2*(b*d - 6*a*e)) + A*e*(-44*c^2*d^2 - 15*b^2*e^2 + 4*c*e*(11*b*d + 4*a*e)))*Sqrt[a + x*(b + c*x)])/(4*(
c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*(b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9*A*c*d*e + 3*a*B*e^2)
+ 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) + 8*c*(a*B*e*(-4*c*d^2 + a*e^2) + A*c*d*(-2*c*d^2
+ 3*a*e^2)))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x
)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1535\) vs. \(2(422)=844\).
time = 0.08, size = 1536, normalized size = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1536\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

B/e^4*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)-3/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a
*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(
(2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*
(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+(A*e-B*d)/e^5*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-5/6*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/2/(a*e^2
-b*d*e+c*d^2)*e^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-3/4*(b*e-2*c*d)*
e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/
e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*
d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2))/(x+d/e)))-2/3*c/(a*e^2-b*d*e+c*d^2)*e^2*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c
+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2
*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))

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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((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1813 vs. \(2 (435) = 870\).
time = 152.27, size = 3669, normalized size = 8.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/96*(3*(8*(B*b*c^2 - 2*A*c^3)*d^6 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*x^3*e^6 + ((B*b^3 + 24*
A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d*x^3 - 3*(6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d*x^2)*e^5 - (4*
(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^2*x^3 - 3*(B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^2*x^2 + 3*(6*B
*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^2*x)*e^4 + (8*(B*b*c^2 - 2*A*c^3)*d^3*x^3 - 12*(B*b^2*c + 2*(4*B
*a - 3*A*b)*c^2)*d^3*x^2 + 3*(B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^3*x - (6*B*a*b^2 - 5*A*b^3 - 4*
(2*B*a^2 - 3*A*a*b)*c)*d^3)*e^3 + (24*(B*b*c^2 - 2*A*c^3)*d^4*x^2 - 12*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^4*x
+ (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^4)*e^2 + 4*(6*(B*b*c^2 - 2*A*c^3)*d^5*x - (B*b^2*c + 2*(4
*B*a - 3*A*b)*c^2)*d^5)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*a*c)*d^2 +
4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4*a*c
)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) - 4*(24*B*c
^3*d^7 - (8*A*a^3 - (18*B*a^2*b - 15*A*a*b^2 + 16*A*a^2*c)*x^2 + 2*(6*B*a^3 - 5*A*a^2*b)*x)*e^7 - ((21*B*a*b^2
- 15*A*b^3 + 4*(13*B*a^2 - 7*A*a*b)*c)*d*x^2 - 2*(31*B*a^2*b - 25*A*a*b^2 + 6*A*a^2*c)*d*x + 2*(2*B*a^3 - 17*
A*a^2*b)*d)*e^6 + ((3*B*b^3 - 28*A*a*c^2 + (80*B*a*b - 59*A*b^2)*c)*d^2*x^2 - 2*(29*B*a*b^2 - 20*A*b^3 + 2*(30
*B*a^2 - 29*A*a*b)*c)*d^2*x + (20*B*a^2*b - 59*A*a*b^2 - 28*A*a^2*c)*d^2)*e^5 - ((13*B*b^2*c + 44*(B*a - 2*A*b
)*c^2)*d^3*x^2 - 2*(4*B*b^3 - 48*A*a*c^2 + (86*B*a*b - 79*A*b^2)*c)*d^3*x + (13*B*a*b^2 - 33*A*b^3 + 4*(11*B*a
^2 - 34*A*a*b)*c)*d^3)*e^4 + (2*(B*b*c^2 - 22*A*c^3)*d^4*x^2 - 2*(11*B*b^2*c + (42*B*a - 113*A*b)*c^2)*d^4*x -
(3*B*b^3 + 92*A*a*c^2 - (44*B*a*b - 123*A*b^2)*c)*d^4)*e^3 + (8*B*c^3*d^5*x^2 - 2*(5*B*b*c^2 + 54*A*c^3)*d^5*
x + (15*B*b^2*c - 2*(8*B*a - 81*A*b)*c^2)*d^5)*e^2 + 12*(2*B*c^3*d^6*x - 3*(B*b*c^2 + 2*A*c^3)*d^6)*e)*sqrt(c*
x^2 + b*x + a))/(c^4*d^11 + a^4*x^3*e^11 - (4*a^3*b*d*x^3 - 3*a^4*d*x^2)*e^10 - (12*a^3*b*d^2*x^2 - 3*a^4*d^2*
x - 2*(3*a^2*b^2 + 2*a^3*c)*d^2*x^3)*e^9 - (12*a^3*b*d^3*x - a^4*d^3 + 4*(a*b^3 + 3*a^2*b*c)*d^3*x^3 - 6*(3*a^
2*b^2 + 2*a^3*c)*d^3*x^2)*e^8 - (4*a^3*b*d^4 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*x^3 + 12*(a*b^3 + 3*a^2*b*c)
*d^4*x^2 - 6*(3*a^2*b^2 + 2*a^3*c)*d^4*x)*e^7 - (4*(b^3*c + 3*a*b*c^2)*d^5*x^3 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c
^2)*d^5*x^2 + 12*(a*b^3 + 3*a^2*b*c)*d^5*x - 2*(3*a^2*b^2 + 2*a^3*c)*d^5)*e^6 + (2*(3*b^2*c^2 + 2*a*c^3)*d^6*x
^3 - 12*(b^3*c + 3*a*b*c^2)*d^6*x^2 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*x - 4*(a*b^3 + 3*a^2*b*c)*d^6)*e^5
- (4*b*c^3*d^7*x^3 - 6*(3*b^2*c^2 + 2*a*c^3)*d^7*x^2 + 12*(b^3*c + 3*a*b*c^2)*d^7*x - (b^4 + 12*a*b^2*c + 6*a^
2*c^2)*d^7)*e^4 + (c^4*d^8*x^3 - 12*b*c^3*d^8*x^2 + 6*(3*b^2*c^2 + 2*a*c^3)*d^8*x - 4*(b^3*c + 3*a*b*c^2)*d^8)
*e^3 + (3*c^4*d^9*x^2 - 12*b*c^3*d^9*x + 2*(3*b^2*c^2 + 2*a*c^3)*d^9)*e^2 + (3*c^4*d^10*x - 4*b*c^3*d^10)*e),
-1/48*(3*(8*(B*b*c^2 - 2*A*c^3)*d^6 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*x^3*e^6 + ((B*b^3 + 24*A
*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d*x^3 - 3*(6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d*x^2)*e^5 - (4*(
B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^2*x^3 - 3*(B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^2*x^2 + 3*(6*B*
a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^2*x)*e^4 + (8*(B*b*c^2 - 2*A*c^3)*d^3*x^3 - 12*(B*b^2*c + 2*(4*B*
a - 3*A*b)*c^2)*d^3*x^2 + 3*(B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^3*x - (6*B*a*b^2 - 5*A*b^3 - 4*(
2*B*a^2 - 3*A*a*b)*c)*d^3)*e^3 + (24*(B*b*c^2 - 2*A*c^3)*d^4*x^2 - 12*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^4*x
+ (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^4)*e^2 + 4*(6*(B*b*c^2 - 2*A*c^3)*d^5*x - (B*b^2*c + 2*(4*
B*a - 3*A*b)*c^2)*d^5)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d
- (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*
c*d*x^2 + b^2*d*x + a*b*d)*e)) - 2*(24*B*c^3*d^7 - (8*A*a^3 - (18*B*a^2*b - 15*A*a*b^2 + 16*A*a^2*c)*x^2 + 2*(
6*B*a^3 - 5*A*a^2*b)*x)*e^7 - ((21*B*a*b^2 - 15*A*b^3 + 4*(13*B*a^2 - 7*A*a*b)*c)*d*x^2 - 2*(31*B*a^2*b - 25*A
*a*b^2 + 6*A*a^2*c)*d*x + 2*(2*B*a^3 - 17*A*a^2*b)*d)*e^6 + ((3*B*b^3 - 28*A*a*c^2 + (80*B*a*b - 59*A*b^2)*c)*
d^2*x^2 - 2*(29*B*a*b^2 - 20*A*b^3 + 2*(30*B*a^2 - 29*A*a*b)*c)*d^2*x + (20*B*a^2*b - 59*A*a*b^2 - 28*A*a^2*c)
*d^2)*e^5 - ((13*B*b^2*c + 44*(B*a - 2*A*b)*c^2)*d^3*x^2 - 2*(4*B*b^3 - 48*A*a*c^2 + (86*B*a*b - 79*A*b^2)*c)*
d^3*x + (13*B*a*b^2 - 33*A*b^3 + 4*(11*B*a^2 - 34*A*a*b)*c)*d^3)*e^4 + (2*(B*b*c^2 - 22*A*c^3)*d^4*x^2 - 2*(11
*B*b^2*c + (42*B*a - 113*A*b)*c^2)*d^4*x - (3*B*b^3 + 92*A*a*c^2 - (44*B*a*b - 123*A*b^2)*c)*d^4)*e^3 + (8*B*c
^3*d^5*x^2 - 2*(5*B*b*c^2 + 54*A*c^3)*d^5*x + (15*B*b^2*c - 2*(8*B*a - 81*A*b)*c^2)*d^5)*e^2 + 12*(2*B*c^3*d^6
*x - 3*(B*b*c^2 + 2*A*c^3)*d^6)*e)*sqrt(c*x^2 + b*x + a))/(c^4*d^11 + a^4*x^3*e^11 - (4*a^3*b*d*x^3 - 3*a^4*d*
x^2)*e^10 - (12*a^3*b*d^2*x^2 - 3*a^4*d^2*x - 2...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (d + e x\right )^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((A + B*x)/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4249 vs. \(2 (435) = 870\).
time = 3.89, size = 4249, normalized size = 9.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/8*(8*B*b*c^2*d^3 - 16*A*c^3*d^3 - 4*B*b^2*c*d^2*e - 32*B*a*c^2*d^2*e + 24*A*b*c^2*d^2*e + B*b^3*d*e^2 + 20*
B*a*b*c*d*e^2 - 18*A*b^2*c*d*e^2 + 24*A*a*c^2*d*e^2 - 6*B*a*b^2*e^3 + 5*A*b^3*e^3 + 8*B*a^2*c*e^3 - 12*A*a*b*c
*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*
b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^
2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/24*(64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*B*c^4*d^6 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b*c^3*d^5*e - 352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*A*c^4*d^5*e + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b*c^(7/2)*d^6 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*B*b*c^(5/2)*d^4*e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*c^(7/2)*d^4*e^2 + 24*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*B*b^2*c^(5/2)*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*c^(7/2)*d^5*e - 528
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b*c^(7/2)*d^5*e + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^2*c^3*d^
6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b*c^2*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*c^3*
d^3*e^3 + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^2*c^2*d^4*e^2 - 512*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^3*B*a*c^3*d^4*e^2 + 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b*c^3*d^4*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*B*b^3*c^2*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b*c^3*d^5*e - 264*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*A*b^2*c^3*d^5*e + 8*B*b^3*c^(5/2)*d^6 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*b^2*c^(3/2
)*d^3*e^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*c^(5/2)*d^3*e^3 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*A*b*c^(5/2)*d^3*e^3 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^3*c^(3/2)*d^4*e^2 - 696*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*B*a*b*c^(5/2)*d^4*e^2 + 756*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^2*c^(5/2)*d^4
*e^2 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*c^(7/2)*d^4*e^2 + 10*B*b^4*c^(3/2)*d^5*e - 48*B*a*b^2*c^(
5/2)*d^5*e - 44*A*b^3*c^(5/2)*d^5*e - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b^2*c*d^2*e^4 - 96*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*B*a*c^2*d^2*e^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b*c^2*d^2*e^4 - 74*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^3*c*d^3*e^3 - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b*c^2*d^3*
e^3 - 204*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2*c^2*d^3*e^3 + 656*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
A*a*c^3*d^3*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^4*c*d^4*e^2 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a*b^2*c^2*d^4*e^2 + 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^3*c^2*d^4*e^2 + 96*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*B*a^2*c^3*d^4*e^2 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b*c^3*d^4*e^2 + 15*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*B*b^3*sqrt(c)*d^2*e^4 + 300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*b*c^(3/2)*d^2*
e^4 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*b^2*c^(3/2)*d^2*e^4 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*A*a*c^(5/2)*d^2*e^4 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^4*sqrt(c)*d^3*e^3 + 252*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*B*a*b^2*c^(3/2)*d^3*e^3 - 498*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^3*c^(3/2)*d^3*e
^3 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*c^(5/2)*d^3*e^3 - 648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*A*a*b*c^(5/2)*d^3*e^3 - 3*B*b^5*sqrt(c)*d^4*e^2 - 70*B*a*b^3*c^(3/2)*d^4*e^2 + 44*A*b^4*c^(3/2)*d^4*e^2 + 4
8*B*a^2*b*c^(5/2)*d^4*e^2 + 204*A*a*b^2*c^(5/2)*d^4*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b^3*d*e^5
+ 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b*c*d*e^5 - 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^2*c*d*
e^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*c^2*d*e^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^4*d
^2*e^4 + 252*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b^2*c*d^2*e^4 - 34*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*A*b^3*c*d^2*e^4 + 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*c^2*d^2*e^4 - 264*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*A*a*b*c^2*d^2*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^5*d^3*e^3 + 186*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*B*a*b^3*c*d^3*e^3 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^4*c*d^3*e^3 + 864*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*B*a^2*b*c^2*d^3*e^3 - 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2*c^2*d^3*e^3 -
480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c^3*d^3*e^3 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*b^2*s
qrt(c)*d*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*b^3*sqrt(c)*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*B*a^2*c^(3/2)*d*e^5 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a*b*c^(3/2)*d*e^5 - 24*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*B*a*b^3*sqrt(c)*d^2*e^4 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^4*sqrt(c)*d
^2*e^4 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*b*c^(3/2)*d^2*e^4 + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*A*a*b^2*c^(3/2)*d^2*e^4 - 288*(sqrt(c)...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)),x)

[Out]

int((A + B*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)), x)

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