3.12.69 \(\int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx\) [1169]

Optimal. Leaf size=200 \[ -\frac {(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {b^2 (b B d-2 A c d+A b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}} \]

[Out]

1/3*(-A*e+B*d)*(c*x^2+b*x)^(3/2)/d/(-b*e+c*d)/(e*x+d)^3+1/16*b^2*(A*b*e-2*A*c*d+B*b*d)*arctanh(1/2*(b*d+(-b*e+
2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/d^(5/2)/(-b*e+c*d)^(5/2)-1/8*(A*b*e-2*A*c*d+B*b*d)*(b*d+
(-b*e+2*c*d)*x)*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d)^2

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Rubi [A]
time = 0.12, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {820, 734, 738, 212} \begin {gather*} \frac {b^2 (A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}-\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{8 d^2 (d+e x)^2 (c d-b e)^2}+\frac {\left (b x+c x^2\right )^{3/2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((b*B*d - 2*A*c*d + A*b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(d^2*(c*d - b*e)^2*(d + e*x)^2) + (
(B*d - A*e)*(b*x + c*x^2)^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) + (b^2*(b*B*d - 2*A*c*d + A*b*e)*ArcTanh[(b*d +
(2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(16*d^(5/2)*(c*d - b*e)^(5/2))

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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx &=\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {(b B d-2 A c d+A b e) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{2 d (c d-b e)}\\ &=-\frac {(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {\left (b^2 (b B d-2 A c d+A b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{16 d^2 (c d-b e)^2}\\ &=-\frac {(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {\left (b^2 (b B d-2 A c d+A b e)\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{8 d^2 (c d-b e)^2}\\ &=-\frac {(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {b^2 (b B d-2 A c d+A b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 10.29, size = 199, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (8 (-B d+A e) x^{3/2} (b+c x)-\frac {3 (b B d-2 A c d+A b e) (d+e x) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (-b d-2 c d x+b e x)+b^2 (d+e x)^2 \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{d^{3/2} (c d-b e)^{3/2} \sqrt {b+c x}}\right )}{24 d (-c d+b e) \sqrt {x} (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(8*(-(B*d) + A*e)*x^(3/2)*(b + c*x) - (3*(b*B*d - 2*A*c*d + A*b*e)*(d + e*x)*(Sqrt[d]*Sqrt[
c*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x]
)/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(3/2)*(c*d - b*e)^(3/2)*Sqrt[b + c*x])))/(24*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x
)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2079\) vs. \(2(178)=356\).
time = 0.61, size = 2080, normalized size = 10.40

method	result	size
default	\(\text {Expression too large to display}\)	\(2080\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B/e^4*(1/2/d/(b*e-c*d)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/4*e*(b*e-2*
c*d)/d/(b*e-c*d)*(1/d/(b*e-c*d)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)-1/2*e*
(b*e-2*c*d)/d/(b*e-c*d)*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2
/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)+d*(b*e-
c*d)/e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c
*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e)))-2*c/d/(b*e-c*d)*e^2*(1/4*(2*c*(x+d/e)+1/e
*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b
*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)
/e^2)^(1/2))))-1/2*c/d/(b*e-c*d)*e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2
*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^
(1/2)+d*(b*e-c*d)/e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/
e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))))+(A*e-B*d)/e^5*(1/3/d/(b*e-c
*d)*e^2/(x+d/e)^3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2*e*(b*e-2*c*d)/d/(b*e-c*d)*(1
/2/d/(b*e-c*d)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/4*e*(b*e-2*c*d)/d/(
b*e-c*d)*(1/d/(b*e-c*d)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)-1/2*e*(b*e-2*c
*d)/d/(b*e-c*d)*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-
2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)+d*(b*e-c*d)/e^2
/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(x+d/e)
^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e)))-2*c/d/(b*e-c*d)*e^2*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*
c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d
)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1
/2))))-1/2*c/d/(b*e-c*d)*e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln
((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)+d*
(b*e-c*d)/e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/
2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/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)*(c*x^2+b*x)^(1/2)/(e*x+d)^4,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-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (189) = 378\).
time = 4.16, size = 1217, normalized size = 6.08 \begin {gather*} \left [\frac {3 \, {\left (A b^{3} x^{3} e^{4} + {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{4} + {\left (3 \, A b^{3} d x^{2} + {\left (B b^{3} - 2 \, A b^{2} c\right )} d x^{3}\right )} e^{3} + 3 \, {\left (A b^{3} d^{2} x + {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{2} x^{2}\right )} e^{2} + {\left (A b^{3} d^{3} + 3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{3} x\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {2 \, c d x - b x e + b d + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) + 2 \, {\left (8 \, B c^{3} d^{5} x^{2} - 3 \, A b^{3} d x^{2} e^{4} + 2 \, {\left (B b c^{2} + 6 \, A c^{3}\right )} d^{5} x - 3 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{5} - {\left (8 \, A b^{3} d^{2} x + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} d^{2} x^{2}\right )} e^{3} + {\left (3 \, A b^{3} d^{3} + {\left (17 \, B b^{2} c - 8 \, A b c^{2}\right )} d^{3} x^{2} + 2 \, {\left (4 \, B b^{3} + 11 \, A b^{2} c\right )} d^{3} x\right )} e^{2} - {\left (2 \, {\left (11 \, B b c^{2} - 2 \, A c^{3}\right )} d^{4} x^{2} + 2 \, {\left (5 \, B b^{2} c + 13 \, A b c^{2}\right )} d^{4} x - 3 \, {\left (B b^{3} - 3 \, A b^{2} c\right )} d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{48 \, {\left (c^{3} d^{9} - b^{3} d^{3} x^{3} e^{6} + 3 \, {\left (b^{2} c d^{4} x^{3} - b^{3} d^{4} x^{2}\right )} e^{5} - 3 \, {\left (b c^{2} d^{5} x^{3} - 3 \, b^{2} c d^{5} x^{2} + b^{3} d^{5} x\right )} e^{4} + {\left (c^{3} d^{6} x^{3} - 9 \, b c^{2} d^{6} x^{2} + 9 \, b^{2} c d^{6} x - b^{3} d^{6}\right )} e^{3} + 3 \, {\left (c^{3} d^{7} x^{2} - 3 \, b c^{2} d^{7} x + b^{2} c d^{7}\right )} e^{2} + 3 \, {\left (c^{3} d^{8} x - b c^{2} d^{8}\right )} e\right )}}, \frac {3 \, {\left (A b^{3} x^{3} e^{4} + {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{4} + {\left (3 \, A b^{3} d x^{2} + {\left (B b^{3} - 2 \, A b^{2} c\right )} d x^{3}\right )} e^{3} + 3 \, {\left (A b^{3} d^{2} x + {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{2} x^{2}\right )} e^{2} + {\left (A b^{3} d^{3} + 3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} d^{3} x\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + {\left (8 \, B c^{3} d^{5} x^{2} - 3 \, A b^{3} d x^{2} e^{4} + 2 \, {\left (B b c^{2} + 6 \, A c^{3}\right )} d^{5} x - 3 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{5} - {\left (8 \, A b^{3} d^{2} x + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} d^{2} x^{2}\right )} e^{3} + {\left (3 \, A b^{3} d^{3} + {\left (17 \, B b^{2} c - 8 \, A b c^{2}\right )} d^{3} x^{2} + 2 \, {\left (4 \, B b^{3} + 11 \, A b^{2} c\right )} d^{3} x\right )} e^{2} - {\left (2 \, {\left (11 \, B b c^{2} - 2 \, A c^{3}\right )} d^{4} x^{2} + 2 \, {\left (5 \, B b^{2} c + 13 \, A b c^{2}\right )} d^{4} x - 3 \, {\left (B b^{3} - 3 \, A b^{2} c\right )} d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{24 \, {\left (c^{3} d^{9} - b^{3} d^{3} x^{3} e^{6} + 3 \, {\left (b^{2} c d^{4} x^{3} - b^{3} d^{4} x^{2}\right )} e^{5} - 3 \, {\left (b c^{2} d^{5} x^{3} - 3 \, b^{2} c d^{5} x^{2} + b^{3} d^{5} x\right )} e^{4} + {\left (c^{3} d^{6} x^{3} - 9 \, b c^{2} d^{6} x^{2} + 9 \, b^{2} c d^{6} x - b^{3} d^{6}\right )} e^{3} + 3 \, {\left (c^{3} d^{7} x^{2} - 3 \, b c^{2} d^{7} x + b^{2} c d^{7}\right )} e^{2} + 3 \, {\left (c^{3} d^{8} x - b c^{2} d^{8}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*(A*b^3*x^3*e^4 + (B*b^3 - 2*A*b^2*c)*d^4 + (3*A*b^3*d*x^2 + (B*b^3 - 2*A*b^2*c)*d*x^3)*e^3 + 3*(A*b^3
*d^2*x + (B*b^3 - 2*A*b^2*c)*d^2*x^2)*e^2 + (A*b^3*d^3 + 3*(B*b^3 - 2*A*b^2*c)*d^3*x)*e)*sqrt(c*d^2 - b*d*e)*l
og((2*c*d*x - b*x*e + b*d + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d)) + 2*(8*B*c^3*d^5*x^2 - 3*A*b^3
*d*x^2*e^4 + 2*(B*b*c^2 + 6*A*c^3)*d^5*x - 3*(B*b^2*c - 2*A*b*c^2)*d^5 - (8*A*b^3*d^2*x + (3*B*b^3 - 7*A*b^2*c
)*d^2*x^2)*e^3 + (3*A*b^3*d^3 + (17*B*b^2*c - 8*A*b*c^2)*d^3*x^2 + 2*(4*B*b^3 + 11*A*b^2*c)*d^3*x)*e^2 - (2*(1
1*B*b*c^2 - 2*A*c^3)*d^4*x^2 + 2*(5*B*b^2*c + 13*A*b*c^2)*d^4*x - 3*(B*b^3 - 3*A*b^2*c)*d^4)*e)*sqrt(c*x^2 + b
*x))/(c^3*d^9 - b^3*d^3*x^3*e^6 + 3*(b^2*c*d^4*x^3 - b^3*d^4*x^2)*e^5 - 3*(b*c^2*d^5*x^3 - 3*b^2*c*d^5*x^2 + b
^3*d^5*x)*e^4 + (c^3*d^6*x^3 - 9*b*c^2*d^6*x^2 + 9*b^2*c*d^6*x - b^3*d^6)*e^3 + 3*(c^3*d^7*x^2 - 3*b*c^2*d^7*x
+ b^2*c*d^7)*e^2 + 3*(c^3*d^8*x - b*c^2*d^8)*e), 1/24*(3*(A*b^3*x^3*e^4 + (B*b^3 - 2*A*b^2*c)*d^4 + (3*A*b^3*
d*x^2 + (B*b^3 - 2*A*b^2*c)*d*x^3)*e^3 + 3*(A*b^3*d^2*x + (B*b^3 - 2*A*b^2*c)*d^2*x^2)*e^2 + (A*b^3*d^3 + 3*(B
*b^3 - 2*A*b^2*c)*d^3*x)*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e
)) + (8*B*c^3*d^5*x^2 - 3*A*b^3*d*x^2*e^4 + 2*(B*b*c^2 + 6*A*c^3)*d^5*x - 3*(B*b^2*c - 2*A*b*c^2)*d^5 - (8*A*b
^3*d^2*x + (3*B*b^3 - 7*A*b^2*c)*d^2*x^2)*e^3 + (3*A*b^3*d^3 + (17*B*b^2*c - 8*A*b*c^2)*d^3*x^2 + 2*(4*B*b^3 +
11*A*b^2*c)*d^3*x)*e^2 - (2*(11*B*b*c^2 - 2*A*c^3)*d^4*x^2 + 2*(5*B*b^2*c + 13*A*b*c^2)*d^4*x - 3*(B*b^3 - 3*
A*b^2*c)*d^4)*e)*sqrt(c*x^2 + b*x))/(c^3*d^9 - b^3*d^3*x^3*e^6 + 3*(b^2*c*d^4*x^3 - b^3*d^4*x^2)*e^5 - 3*(b*c^
2*d^5*x^3 - 3*b^2*c*d^5*x^2 + b^3*d^5*x)*e^4 + (c^3*d^6*x^3 - 9*b*c^2*d^6*x^2 + 9*b^2*c*d^6*x - b^3*d^6)*e^3 +
3*(c^3*d^7*x^2 - 3*b*c^2*d^7*x + b^2*c*d^7)*e^2 + 3*(c^3*d^8*x - b*c^2*d^8)*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1555 vs. \(2 (189) = 378\).
time = 3.71, size = 1555, normalized size = 7.78 \begin {gather*} \frac {{\left (B b^{3} d - 2 \, A b^{2} c d + A b^{3} e\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{8 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} + \frac {96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B c^{\frac {7}{2}} d^{5} e + 64 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B c^{4} d^{6} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B c^{3} d^{4} e^{2} - 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c^{3} d^{5} e + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{4} d^{5} e + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{\frac {7}{2}} d^{6} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b c^{\frac {5}{2}} d^{4} e^{2} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A c^{\frac {7}{2}} d^{4} e^{2} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} c^{\frac {5}{2}} d^{5} e + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{\frac {7}{2}} d^{5} e + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{2} c^{3} d^{6} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b c^{2} d^{3} e^{3} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} c^{2} d^{4} e^{2} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{3} d^{4} e^{2} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} c^{2} d^{5} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c^{3} d^{5} e + 8 \, B b^{3} c^{\frac {5}{2}} d^{6} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b c^{\frac {5}{2}} d^{3} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{3} c^{\frac {3}{2}} d^{4} e^{2} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c^{\frac {5}{2}} d^{4} e^{2} - 14 \, B b^{4} c^{\frac {3}{2}} d^{5} e + 4 \, A b^{3} c^{\frac {5}{2}} d^{5} e + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{2} c d^{2} e^{4} + 58 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} c d^{3} e^{3} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{2} d^{3} e^{3} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{4} c d^{4} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} c^{2} d^{4} e^{2} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{3} \sqrt {c} d^{2} e^{4} + 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{2} c^{\frac {3}{2}} d^{2} e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{4} \sqrt {c} d^{3} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c^{\frac {3}{2}} d^{3} e^{3} + 3 \, B b^{5} \sqrt {c} d^{4} e^{2} - 4 \, A b^{4} c^{\frac {3}{2}} d^{4} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} d e^{5} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c d e^{5} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{4} d^{2} e^{4} + 74 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{3} c d^{2} e^{4} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{5} d^{3} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} c d^{3} e^{3} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} \sqrt {c} d e^{5} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{4} \sqrt {c} d^{2} e^{4} + 3 \, A b^{5} \sqrt {c} d^{3} e^{3} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{3} e^{6} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} d e^{5} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{5} d^{2} e^{4}}{24 \, {\left (c^{2} d^{4} e^{3} - 2 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(B*b^3*d - 2*A*b^2*c*d + A*b^3*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*
d*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) + 1/24*(96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
4*B*c^(7/2)*d^5*e + 64*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^4*d^6 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^
3*d^4*e^2 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^3*d^5*e + 32*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^4*d^
5*e + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^(7/2)*d^6 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^(5/2)
*d^4*e^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*c^(7/2)*d^4*e^2 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^
2*c^(5/2)*d^5*e + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^(7/2)*d^5*e + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))*
B*b^2*c^3*d^6 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^2*d^3*e^3 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B
*b^2*c^2*d^4*e^2 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^3*d^4*e^2 - 84*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B
*b^3*c^2*d^5*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^3*d^5*e + 8*B*b^3*c^(5/2)*d^6 - 96*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^4*A*b*c^(5/2)*d^3*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c^(3/2)*d^4*e^2 - 36*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c^(5/2)*d^4*e^2 - 14*B*b^4*c^(3/2)*d^5*e + 4*A*b^3*c^(5/2)*d^5*e + 48*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c*d^2*e^4 + 58*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c*d^3*e^3 - 84*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^2*d^3*e^3 + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c*d^4*e^2 - 24*(sqrt(
c)*x - sqrt(c*x^2 + b*x))*A*b^3*c^2*d^4*e^2 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*sqrt(c)*d^2*e^4 + 78*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(3/2)*d^2*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*sqrt(c)*d
^3*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c^(3/2)*d^3*e^3 + 3*B*b^5*sqrt(c)*d^4*e^2 - 4*A*b^4*c^(3/2)
*d^4*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*d*e^5 + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c*d*e^5
+ 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*d^2*e^4 + 74*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c*d^2*e^4 +
3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*d^3*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*c*d^3*e^3 - 15*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*sqrt(c)*d*e^5 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^4*sqrt(c)*d^2*e^4
+ 3*A*b^5*sqrt(c)*d^3*e^3 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*e^6 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^3*A*b^4*d*e^5 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*d^2*e^4)/((c^2*d^4*e^3 - 2*b*c*d^3*e^4 + b^2*d^2*e^5)
*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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