3.3.100 \(\int \frac {(b x+c x^2)^{3/2}}{(d+e x)^2} \, dx\) [300]

Optimal. Leaf size=198 \[ -\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^4}-\frac {3 \sqrt {d} \sqrt {c d-b e} (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 e^4} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 198, 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, 828, 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 {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^4}-\frac {3 \sqrt {d} \sqrt {c d-b e} (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 )}{2 e^4}-\frac {3 \sqrt {b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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)^2} \, dx &=-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \int \frac {(b+2 c x) \sqrt {b x+c x^2}}{d+e x} \, dx}{2 e}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}-\frac {3 \int \frac {-b c d (4 c d-3 b e)-c \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 c e^3}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}-\frac {(3 d (c d-b e) (2 c d-b e)) \int \frac {1}{(d+e x) \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}{\sqrt {b x+c x^2}} \, dx}{8 e^4}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {(3 d (c d-b e) (2 c d-b e)) \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 )}{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}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 e^4}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^4}-\frac {3 \sqrt {d} \sqrt {c d-b e} (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 )}{2 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 225, normalized size = 1.14 \begin {gather*} \frac {(x (b+c x))^{3/2} \left (\frac {e \sqrt {x} \left (b e (9 d+5 e x)-2 c \left (6 d^2+3 d e x-e^2 x^2\right )\right )}{(b+c x) (d+e x)}-\frac {12 \sqrt {d} (2 c d-b e) \sqrt {-c d+b e} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(b+c x)^{3/2}}-\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {c} (b+c x)^{3/2}}\right )}{4 e^4 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(b*e*(9*d + 5*e*x) - 2*c*(6*d^2 + 3*d*e*x - e^2*x^2)))/((b + c*x)*(d + e*x))
- (12*Sqrt[d]*(2*c*d - b*e)*Sqrt[-(c*d) + b*e]*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d
]*Sqrt[-(c*d) + b*e])])/(b + c*x)^(3/2) - (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b
+ c*x]])/(Sqrt[c]*(b + c*x)^(3/2))))/(4*e^4*x^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(965\) vs. \(2(170)=340\).
time = 0.50, size = 966, normalized size = 4.88

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 {5}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {3 e \left (b e -2 c d \right ) \left (\frac {\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}+\frac {\left (b e -2 c d \right ) \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 )}{2 e}-\frac {d \left (b e -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 )}{e^{2}}\right )}{2 d \left (b e -c d \right )}-\frac {4 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \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}}}{8 c}+\frac {3 \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 ) \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 )}{16 c}\right )}{d \left (b e -c d \right )}}{e^{2}}\)	\(966\)
risch	\(\text {Expression too large to display}\)	\(1361\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^2*(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)))))

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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)^(3/2)/(e*x+d)^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-%e*b>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 2.19, size = 1007, normalized size = 5.09 \begin {gather*} \left [\frac {3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 12 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\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^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c x e^{5} + c d e^{4}\right )}}, -\frac {24 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\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 ) - 3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c x e^{5} + c d e^{4}\right )}}, -\frac {3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + 6 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\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 ) + {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c x e^{5} + c d e^{4}\right )}}, -\frac {12 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\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 ) + 3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c x e^{5} + c d e^{4}\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)^2,x, algorithm="fricas")

[Out]

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

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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 )^{2}}\, 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)**2,x)

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,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)^2,x)

[Out]

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

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