3.3.91 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx\) [291]

Optimal. Leaf size=127 \[ \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \]

[Out]

-1/8*b^2*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^(3/2)/(-b*e+c*d)^(3/2)
+1/4*(b*d+(-b*e+2*c*d)*x)*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^2

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Rubi [A]
time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {734, 738, 212} \begin {gather*} \frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \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 )}{8 d^{3/2} (c d-b e)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^3,x]

[Out]

((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2*ArcTanh[(b*d + (2*c*d - b*e)*
x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(3/2)*(c*d - b*e)^(3/2))

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx &=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d (c d-b e)}\\ &=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}+\frac {b^2 \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 )}{4 d (c d-b e)}\\ &=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \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 )}{8 d^{3/2} (c d-b e)^{3/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1006\) vs. \(2(127)=254\).
time = 6.70, size = 1006, normalized size = 7.92 \begin {gather*} \frac {-\frac {b (b+c x) \left (3 b \sqrt {c} \sqrt {x}+4 c^{3/2} x^{3/2}-b \sqrt {b+c x}-4 c x \sqrt {b+c x}\right )}{d e^2 \left (b^2+8 b c x+8 c^2 x^2-4 b \sqrt {c} \sqrt {x} \sqrt {b+c x}-8 c^{3/2} x^{3/2} \sqrt {b+c x}\right )}-\frac {b c^{3/2} \sqrt {x} \left (b^2+5 b c x+4 c^2 x^2-3 b \sqrt {c} \sqrt {x} \sqrt {b+c x}-4 c^{3/2} x^{3/2} \sqrt {b+c x}\right )}{e^2 (-c d+b e) \left (b^2+8 b c x+8 c^2 x^2-4 b \sqrt {c} \sqrt {x} \sqrt {b+c x}-8 c^{3/2} x^{3/2} \sqrt {b+c x}\right )}+\frac {2 x (b+c x) \left (8 c^{5/2} x^{5/2}-8 c^2 x^2 \sqrt {b+c x}-b^2 \left (-4 \sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )+b \left (12 c^{3/2} x^{3/2}-8 c x \sqrt {b+c x}\right )\right )}{e (d+e x)^2 \left (b+2 c x-2 \sqrt {c} \sqrt {x} \sqrt {b+c x}\right )^2}+\frac {(b+2 c x) \left (8 c^{5/2} x^{5/2}-8 c^2 x^2 \sqrt {b+c x}-b^2 \left (-4 \sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )+b \left (12 c^{3/2} x^{3/2}-8 c x \sqrt {b+c x}\right )\right )}{e^2 (d+e x) \left (b^2+8 b c x+8 c^2 x^2-4 b \sqrt {c} \sqrt {x} \sqrt {b+c x}-8 c^{3/2} x^{3/2} \sqrt {b+c x}\right )}+\frac {8 c^2 \sqrt {d} \sqrt {x} (b+c x) \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{e^2 (-c d+b e)^{3/2}}-\frac {8 b c \sqrt {x} (b+c x) \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{\sqrt {d} e (-c d+b e)^{3/2}}-\frac {b^2 \sqrt {x} (b+c x) \tanh ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac {8 c^2 \sqrt {d} \sqrt {x} (b+c x) \tanh ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^2 (c d-b e)^{3/2}}+\frac {8 b c \sqrt {x} (b+c x) \tanh ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {c d-b e}}\right )}{\sqrt {d} e (c d-b e)^{3/2}}}{4 \sqrt {b+c x} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*(b + c*x)*(3*b*Sqrt[c]*Sqrt[x] + 4*c^(3/2)*x^(3/2) - b*Sqrt[b + c*x] - 4*c*x*Sqrt[b + c*x]))/(d*e^2*(b^2
+ 8*b*c*x + 8*c^2*x^2 - 4*b*Sqrt[c]*Sqrt[x]*Sqrt[b + c*x] - 8*c^(3/2)*x^(3/2)*Sqrt[b + c*x]))) - (b*c^(3/2)*S
qrt[x]*(b^2 + 5*b*c*x + 4*c^2*x^2 - 3*b*Sqrt[c]*Sqrt[x]*Sqrt[b + c*x] - 4*c^(3/2)*x^(3/2)*Sqrt[b + c*x]))/(e^2
*(-(c*d) + b*e)*(b^2 + 8*b*c*x + 8*c^2*x^2 - 4*b*Sqrt[c]*Sqrt[x]*Sqrt[b + c*x] - 8*c^(3/2)*x^(3/2)*Sqrt[b + c*
x])) + (2*x*(b + c*x)*(8*c^(5/2)*x^(5/2) - 8*c^2*x^2*Sqrt[b + c*x] - b^2*(-4*Sqrt[c]*Sqrt[x] + Sqrt[b + c*x])
+ b*(12*c^(3/2)*x^(3/2) - 8*c*x*Sqrt[b + c*x])))/(e*(d + e*x)^2*(b + 2*c*x - 2*Sqrt[c]*Sqrt[x]*Sqrt[b + c*x])^
2) + ((b + 2*c*x)*(8*c^(5/2)*x^(5/2) - 8*c^2*x^2*Sqrt[b + c*x] - b^2*(-4*Sqrt[c]*Sqrt[x] + Sqrt[b + c*x]) + b*
(12*c^(3/2)*x^(3/2) - 8*c*x*Sqrt[b + c*x])))/(e^2*(d + e*x)*(b^2 + 8*b*c*x + 8*c^2*x^2 - 4*b*Sqrt[c]*Sqrt[x]*S
qrt[b + c*x] - 8*c^(3/2)*x^(3/2)*Sqrt[b + c*x])) + (8*c^2*Sqrt[d]*Sqrt[x]*(b + c*x)*ArcTan[(-(e*Sqrt[x]*Sqrt[b
+ c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(e^2*(-(c*d) + b*e)^(3/2)) - (8*b*c*Sqrt[x]*(b +
c*x)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(Sqrt[d]*e*(-(c*d)
+ b*e)^(3/2)) - (b^2*Sqrt[x]*(b + c*x)*ArcTanh[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt
[c*d - b*e])])/(d^(3/2)*(c*d - b*e)^(3/2)) - (8*c^2*Sqrt[d]*Sqrt[x]*(b + c*x)*ArcTanh[(-(e*Sqrt[x]*Sqrt[b + c*
x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[c*d - b*e])])/(e^2*(c*d - b*e)^(3/2)) + (8*b*c*Sqrt[x]*(b + c*x)*ArcTan
h[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[c*d - b*e])])/(Sqrt[d]*e*(c*d - b*e)^(3/2)))/
(4*Sqrt[b + c*x]*Sqrt[x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(109)=218\).
time = 0.47, size = 986, normalized size = 7.76

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}}}{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 )}}{e^{3}}\)	\(986\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^3*(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)^3,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. 236 vs. \(2 (115) = 230\).
time = 2.58, size = 484, normalized size = 3.81 \begin {gather*} \left [-\frac {{\left (b^{2} x^{2} e^{2} + 2 \, b^{2} d x e + b^{2} d^{2}\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 (2 \, c^{2} d^{3} x + b c d^{3} + b^{2} d x e^{2} - {\left (3 \, b c d^{2} x + b^{2} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{2} d^{6} + b^{2} d^{2} x^{2} e^{4} - 2 \, {\left (b c d^{3} x^{2} - b^{2} d^{3} x\right )} e^{3} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + b^{2} d^{4}\right )} e^{2} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e\right )}}, -\frac {{\left (b^{2} x^{2} e^{2} + 2 \, b^{2} d x e + b^{2} d^{2}\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 (2 \, c^{2} d^{3} x + b c d^{3} + b^{2} d x e^{2} - {\left (3 \, b c d^{2} x + b^{2} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{2} d^{6} + b^{2} d^{2} x^{2} e^{4} - 2 \, {\left (b c d^{3} x^{2} - b^{2} d^{3} x\right )} e^{3} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + b^{2} d^{4}\right )} e^{2} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\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)^3,x, algorithm="fricas")

[Out]

[-1/8*((b^2*x^2*e^2 + 2*b^2*d*x*e + b^2*d^2)*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*(2*c^2*d^3*x + b*c*d^3 + b^2*d*x*e^2 - (3*b*c*d^2*x + b^2*d^2)*e)*sqrt
(c*x^2 + b*x))/(c^2*d^6 + b^2*d^2*x^2*e^4 - 2*(b*c*d^3*x^2 - b^2*d^3*x)*e^3 + (c^2*d^4*x^2 - 4*b*c*d^4*x + b^2
*d^4)*e^2 + 2*(c^2*d^5*x - b*c*d^5)*e), -1/4*((b^2*x^2*e^2 + 2*b^2*d*x*e + b^2*d^2)*sqrt(-c*d^2 + b*d*e)*arcta
n(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) - (2*c^2*d^3*x + b*c*d^3 + b^2*d*x*e^2 - (3*b*c*d^2
*x + b^2*d^2)*e)*sqrt(c*x^2 + b*x))/(c^2*d^6 + b^2*d^2*x^2*e^4 - 2*(b*c*d^3*x^2 - b^2*d^3*x)*e^3 + (c^2*d^4*x^
2 - 4*b*c*d^4*x + b^2*d^4)*e^2 + 2*(c^2*d^5*x - b*c*d^5)*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 )^{3}}\, 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)**3,x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (115) = 230\).
time = 2.48, size = 409, normalized size = 3.22 \begin {gather*} -\frac {b^{2} \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 )}{4 \, {\left (c d^{2} - b d e\right )} \sqrt {-c d^{2} + b d e}} + \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 2 \, b^{2} c^{\frac {3}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} - b^{3} \sqrt {c} d^{2} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2}}{4 \, {\left (c d^{2} e^{2} - b d e^{3}\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 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

-1/4*b^2*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c*d^2 - b*d*e)*sqrt(-
c*d^2 + b*d*e)) + 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^2*d^2*e + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(
5/2)*d^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 - 4*(sq
rt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2*e + 2*b^2*c^(3/2)*d^3 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c)
*d*e^2 - b^3*sqrt(c)*d^2*e + (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 - (sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*d
*e^2)/((c*d^2*e^2 - b*d*e^3)*((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)^2)

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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 )}^3} \,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)^3,x)

[Out]

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

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