3.4.1 \(\int \frac {(b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [301]

Optimal. Leaf size=205 \[ \frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 \sqrt {d} e^4 \sqrt {c d-b e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {746, 826, 857, 634, 212, 738} \begin {gather*} \frac {3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \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 \sqrt {d} e^4 \sqrt {c d-b e}}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 826

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

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 10.99, size = 259, normalized size = 1.26 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {e \sqrt {x} \left (-b^2 e^2 (3 d+5 e x)-2 c^2 d \left (6 d^2+9 d e x+2 e^2 x^2\right )+b c e \left (15 d^2+23 d e x+4 e^2 x^2\right )\right )}{(d+e x)^2}+\frac {12 \sqrt {c} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}-\frac {3 \sqrt {c d-b e} \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {d} \sqrt {b+c x}}\right )}{4 e^4 (-c d+b e) \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(-(b^2*e^2*(3*d + 5*e*x)) - 2*c^2*d*(6*d^2 + 9*d*e*x + 2*e^2*x^2) + b*c*e*(15*d
^2 + 23*d*e*x + 4*e^2*x^2)))/(d + e*x)^2 + (12*Sqrt[c]*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*ArcSinh[(Sqrt[c]*Sqrt
[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) - (3*Sqrt[c*d - b*e]*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTanh[(Sqr
t[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[b + c*x])))/(4*e^4*(-(c*d) + b*e)*Sqrt[x])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1627\) vs. \(2(177)=354\).
time = 0.53, size = 1628, normalized size = 7.94

method	result	size
default	\(\text {Expression too large to display}\)	\(1628\)
risch	\(\text {Expression too large to display}\)	\(2570\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(3/2)/(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)^(5/2)-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)^(5/2)-3/2*e*
(b*e-2*c*d)/d/(b*e-c*d)*(1/3*(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)*(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)))-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))))-4*c/d/(b*e-c*d)*
e^2*(1/8*(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)^(3/2)+3/16*(-4*
c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(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)))))-3/2*c/d/(b*e-c*d)*e^2*(1/3*(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)*(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)))-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)))))

________________________________________________________________________________________

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)^(3/2)/(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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (181) = 362\).
time = 2.15, size = 1711, normalized size = 8.35 \begin {gather*} \left [-\frac {12 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c 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^{2} d^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}, \frac {3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c 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 ) - 6 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + {\left (12 \, c^{2} d^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}, \frac {24 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + 3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c 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^{2} d^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}, \frac {3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c 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 ) + 12 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (12 \, c^{2} d^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(12*(2*c^2*d^5 + b^2*d*x^2*e^4 - (3*b*c*d^2*x^2 - 2*b^2*d^2*x)*e^3 + (2*c^2*d^3*x^2 - 6*b*c*d^3*x + b^2*
d^3)*e^2 + (4*c^2*d^4*x - 3*b*c*d^4)*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 3*(8*c^2*d^4 +
b^2*x^2*e^4 - 2*(4*b*c*d*x^2 - b^2*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2*x + b^2*d^2)*e^2 + 8*(2*c^2*d^3*x -
b*c*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^2*d^4*e - (4*b*c*d*x^2 - 5*b^2*d*x)*e^4 + (4*c^2*d^2*x^2 - 23*b*c*d^2*x + 3*b^2*d^2)*e^3 + 3*(6*c
^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d*x^2*e^7 + (c*d^2*x^2 - 2*b*d^2*x)*e^6 + (2*c*d^
3*x - b*d^3)*e^5), 1/4*(3*(8*c^2*d^4 + b^2*x^2*e^4 - 2*(4*b*c*d*x^2 - b^2*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d
^2*x + b^2*d^2)*e^2 + 8*(2*c^2*d^3*x - b*c*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)) - 6*(2*c^2*d^5 + b^2*d*x^2*e^4 - (3*b*c*d^2*x^2 - 2*b^2*d^2*x)*e^3 + (2*c^2*d^3*x^2
- 6*b*c*d^3*x + b^2*d^3)*e^2 + (4*c^2*d^4*x - 3*b*c*d^4)*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c
)) + (12*c^2*d^4*e - (4*b*c*d*x^2 - 5*b^2*d*x)*e^4 + (4*c^2*d^2*x^2 - 23*b*c*d^2*x + 3*b^2*d^2)*e^3 + 3*(6*c^2
*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d*x^2*e^7 + (c*d^2*x^2 - 2*b*d^2*x)*e^6 + (2*c*d^3*
x - b*d^3)*e^5), 1/8*(24*(2*c^2*d^5 + b^2*d*x^2*e^4 - (3*b*c*d^2*x^2 - 2*b^2*d^2*x)*e^3 + (2*c^2*d^3*x^2 - 6*b
*c*d^3*x + b^2*d^3)*e^2 + (4*c^2*d^4*x - 3*b*c*d^4)*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + 3*(
8*c^2*d^4 + b^2*x^2*e^4 - 2*(4*b*c*d*x^2 - b^2*d*x)*e^3 + (8*c^2*d^2*x^2 - 16*b*c*d^2*x + b^2*d^2)*e^2 + 8*(2*
c^2*d^3*x - b*c*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^2*d^4*e - (4*b*c*d*x^2 - 5*b^2*d*x)*e^4 + (4*c^2*d^2*x^2 - 23*b*c*d^2*x + 3*b^2*d^2)*
e^3 + 3*(6*c^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d*x^2*e^7 + (c*d^2*x^2 - 2*b*d^2*x)*e
^6 + (2*c*d^3*x - b*d^3)*e^5), 1/4*(3*(8*c^2*d^4 + b^2*x^2*e^4 - 2*(4*b*c*d*x^2 - b^2*d*x)*e^3 + (8*c^2*d^2*x^
2 - 16*b*c*d^2*x + b^2*d^2)*e^2 + 8*(2*c^2*d^3*x - b*c*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*(2*c^2*d^5 + b^2*d*x^2*e^4 - (3*b*c*d^2*x^2 - 2*b^2*d^2*x)*e^3 + (2
*c^2*d^3*x^2 - 6*b*c*d^3*x + b^2*d^3)*e^2 + (4*c^2*d^4*x - 3*b*c*d^4)*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqr
t(-c)/(c*x)) + (12*c^2*d^4*e - (4*b*c*d*x^2 - 5*b^2*d*x)*e^4 + (4*c^2*d^2*x^2 - 23*b*c*d^2*x + 3*b^2*d^2)*e^3
+ 3*(6*c^2*d^3*x - 5*b*c*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d*x^2*e^7 + (c*d^2*x^2 - 2*b*d^2*x)*e^6 +
(2*c*d^3*x - b*d^3)*e^5)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\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)**(3/2)/(e*x+d)**3,x)

[Out]

Integral((x*(b + c*x))**(3/2)/(d + e*x)**3, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (181) = 362\).
time = 2.86, size = 500, normalized size = 2.44 \begin {gather*} \sqrt {c x^{2} + b x} c e^{\left (-3\right )} + \frac {3 \, {\left (8 \, c^{2} d^{2} - 8 \, b c d e + b^{2} e^{2}\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 ) e^{\left (-4\right )}}{4 \, \sqrt {-c d^{2} + b d e}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c}} + \frac {{\left (24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 10 \, b^{2} c^{\frac {3}{2}} d^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 5 \, b^{3} \sqrt {c} d^{2} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2}\right )} e^{\left (-4\right )}}{4 \, {\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)^(3/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

sqrt(c*x^2 + b*x)*c*e^(-3) + 3/4*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e
+ sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^(-4)/sqrt(-c*d^2 + b*d*e) + 3/2*(2*c^2*d - b*c*e)*e^(-4)*log(abs(-2*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/sqrt(c) + 1/4*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^2*d^2*e + 40*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2)*d^2*e + 40*(sqrt(
c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 - 28*(sqrt(c)*x - sqrt(c*
x^2 + b*x))*b^2*c*d^2*e + 10*b^2*c^(3/2)*d^3 - (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c)*d*e^2 - 5*b^3*sqr
t(c)*d^2*e + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*d*e^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)^2

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________