3.3.92 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx\) [292]

Optimal. Leaf size=183 \[ \frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}} \]

[Out]

-1/3*e*(c*x^2+b*x)^(3/2)/d/(-b*e+c*d)/(e*x+d)^3-1/16*b^2*(-b*e+2*c*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*(-b*e+2*c*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 = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {744, 734, 738, 212} \begin {gather*} -\frac {b^2 (2 c d-b e) \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} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]

[Out]

((2*c*d - 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) - (e*(b*x + c*x^2)
^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - (b^2*(2*c*d - 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 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(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[(2*c*d - b*e)/(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, m, p}, 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 + 3, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx &=-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {(2 c d-b e) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{2 d (c d-b e)}\\ &=\frac {(2 c d-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 {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {\left (b^2 (2 c d-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 {(2 c d-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 {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {\left (b^2 (2 c d-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 {(2 c d-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 {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-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 = 10.24, size = 191, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {e x^{3/2} (b+c x)}{(d+e x)^3}-\frac {3 (2 c d-b e) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (2 c d x+b (d-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 )}{8 d^{3/2} (c d-b e)^{3/2} \sqrt {b+c x} (d+e x)^2}\right )}{3 d (-c d+b e) \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1084\) vs. \(2(161)=322\).
time = 0.48, size = 1085, normalized size = 5.93

method	result	size
default	\(\frac {\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {e \left (b e -2 c d \right ) \left (\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {e \left (b e -2 c d \right ) \left (\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {e \left (b e -2 c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{d \left (b e -c d \right )}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}\right )}{2 d \left (b e -c d \right )}}{e^{4}}\)	\(1085\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^4*(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((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. 473 vs. \(2 (172) = 344\).
time = 1.19, size = 958, normalized size = 5.23 \begin {gather*} \left [-\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} x^{3} e^{4} + {\left (2 \, b^{2} c d x^{3} - 3 \, b^{3} d x^{2}\right )} e^{3} + 3 \, {\left (2 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{2} + {\left (6 \, b^{2} c d^{3} x - b^{3} d^{3}\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 (12 \, c^{3} d^{5} x + 6 \, b c^{2} d^{5} - 3 \, b^{3} d x^{2} e^{4} + {\left (7 \, b^{2} c d^{2} x^{2} - 8 \, b^{3} d^{2} x\right )} e^{3} - {\left (8 \, b c^{2} d^{3} x^{2} - 22 \, b^{2} c d^{3} x - 3 \, b^{3} d^{3}\right )} e^{2} + {\left (4 \, c^{3} d^{4} x^{2} - 26 \, b c^{2} d^{4} x - 9 \, b^{2} c 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 (2 \, b^{2} c d^{4} - b^{3} x^{3} e^{4} + {\left (2 \, b^{2} c d x^{3} - 3 \, b^{3} d x^{2}\right )} e^{3} + 3 \, {\left (2 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{2} + {\left (6 \, b^{2} c d^{3} x - b^{3} d^{3}\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 (12 \, c^{3} d^{5} x + 6 \, b c^{2} d^{5} - 3 \, b^{3} d x^{2} e^{4} + {\left (7 \, b^{2} c d^{2} x^{2} - 8 \, b^{3} d^{2} x\right )} e^{3} - {\left (8 \, b c^{2} d^{3} x^{2} - 22 \, b^{2} c d^{3} x - 3 \, b^{3} d^{3}\right )} e^{2} + {\left (4 \, c^{3} d^{4} x^{2} - 26 \, b c^{2} d^{4} x - 9 \, b^{2} c 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((c*x^2+b*x)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/48*(3*(2*b^2*c*d^4 - b^3*x^3*e^4 + (2*b^2*c*d*x^3 - 3*b^3*d*x^2)*e^3 + 3*(2*b^2*c*d^2*x^2 - b^3*d^2*x)*e^2
+ (6*b^2*c*d^3*x - b^3*d^3)*e)*sqrt(c*d^2 - b*d*e)*log((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*(12*c^3*d^5*x + 6*b*c^2*d^5 - 3*b^3*d*x^2*e^4 + (7*b^2*c*d^2*x^2 - 8*b^3*d^2*x)*e^3
- (8*b*c^2*d^3*x^2 - 22*b^2*c*d^3*x - 3*b^3*d^3)*e^2 + (4*c^3*d^4*x^2 - 26*b*c^2*d^4*x - 9*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*(2*b^2*c*d^4 - b^3*x^3*e^4 + (2*b^2*c*d*x
^3 - 3*b^3*d*x^2)*e^3 + 3*(2*b^2*c*d^2*x^2 - b^3*d^2*x)*e^2 + (6*b^2*c*d^3*x - b^3*d^3)*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)) - (12*c^3*d^5*x + 6*b*c^2*d^5 - 3*b^3*d*x^2*
e^4 + (7*b^2*c*d^2*x^2 - 8*b^3*d^2*x)*e^3 - (8*b*c^2*d^3*x^2 - 22*b^2*c*d^3*x - 3*b^3*d^3)*e^2 + (4*c^3*d^4*x^
2 - 26*b*c^2*d^4*x - 9*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 (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(b + c*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. 825 vs. \(2 (172) = 344\).
time = 3.23, size = 825, normalized size = 4.51 \begin {gather*} \frac {{\left (2 \, b^{2} c d - 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 {48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{4} d^{5} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{3} d^{4} e + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {7}{2}} d^{5} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b c^{\frac {5}{2}} d^{3} e^{2} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c^{\frac {5}{2}} d^{4} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c^{3} d^{5} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} c^{2} d^{3} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} c^{2} d^{4} e + 4 \, b^{3} c^{\frac {5}{2}} d^{5} + 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{2} c^{\frac {3}{2}} d^{2} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{3} c^{\frac {3}{2}} d^{3} e^{2} - 4 \, b^{4} c^{\frac {3}{2}} d^{4} e + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{2} c d e^{4} + 74 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{3} c d^{2} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{4} c d^{3} e^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{3} \sqrt {c} d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{4} \sqrt {c} d^{2} e^{3} + 3 \, b^{5} \sqrt {c} d^{3} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{3} e^{5} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{4} d e^{4} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{5} d^{2} e^{3}}{24 \, {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\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((c*x^2+b*x)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/8*(2*b^2*c*d - 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*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^(7/2)*d^4*e
+ 32*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^4*d^5 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c^3*d^4*e + 48*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b*c^(7/2)*d^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b*c^(5/2)*d^3*e^2 - 36*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b^2*c^(5/2)*d^4*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c^3*d^5 - 84*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*b^2*c^2*d^3*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*c^2*d^4*e + 4*b^3*c^(5/2)*d^5
+ 78*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^2*c^(3/2)*d^2*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c^(3/2)*
d^3*e^2 - 4*b^4*c^(3/2)*d^4*e + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^2*c*d*e^4 + 74*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*b^3*c*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^4*c*d^3*e^2 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^4*b^3*sqrt(c)*d*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^4*sqrt(c)*d^2*e^3 + 3*b^5*sqrt(c)*d^3*e^2 - 3*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^5*b^3*e^5 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^4*d*e^4 + 3*(sqrt(c)*x - sqrt(
c*x^2 + b*x))*b^5*d^2*e^3)/((c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4)*((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 = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{{\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)/(d + e*x)^4,x)

[Out]

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

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