3.4.26 \(\int \frac {1}{(d+e x) (b x+c x^2)^{3/2}} \, dx\) [326]

Optimal. Leaf size=126 \[ -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {e^2 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{3/2} (c d-b e)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 126, 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 = {754, 12, 738, 212} \begin {gather*} \frac {e^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 )}{d^{3/2} (c d-b e)^{3/2}}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)^(3/2)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.38, size = 144, normalized size = 1.14 \begin {gather*} \frac {2 \left (\sqrt {d} \sqrt {-c d+b e} \left (-b^2 e+2 c^2 d x+b c (d-e x)\right )+b^2 e^2 \sqrt {x} \sqrt {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 )\right )}{b^2 d^{3/2} (-c d+b e)^{3/2} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(b*x + c*x^2)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(112)=224\).
time = 0.50, size = 340, normalized size = 2.70

method	result	size
risch	\(-\frac {2 \left (c x +b \right )}{b^{2} d \sqrt {x \left (c x +b \right )}}+\frac {e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {2 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{b^{2} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )}\)	\(214\)
default	\(\frac {-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e}\)	\(340\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.67, size = 458, normalized size = 3.63 \begin {gather*} \left [-\frac {{\left (b^{2} c x^{2} + b^{3} x\right )} \sqrt {c d^{2} - b d e} e^{2} \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^{3} d^{3} x + b c^{2} d^{3} + {\left (b^{2} c d x + b^{3} d\right )} e^{2} - {\left (3 \, b c^{2} d^{2} x + 2 \, b^{2} c d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{b^{2} c^{3} d^{4} x^{2} + b^{3} c^{2} d^{4} x + {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e^{2} - 2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x\right )} e}, \frac {2 \, {\left ({\left (b^{2} c x^{2} + b^{3} x\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 ) e^{2} - {\left (2 \, c^{3} d^{3} x + b c^{2} d^{3} + {\left (b^{2} c d x + b^{3} d\right )} e^{2} - {\left (3 \, b c^{2} d^{2} x + 2 \, b^{2} c d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}\right )}}{b^{2} c^{3} d^{4} x^{2} + b^{3} c^{2} d^{4} x + {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e^{2} - 2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((b^2*c*x^2 + b^3*x)*sqrt(c*d^2 - b*d*e)*e^2*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^3*d^3*x + b*c^2*d^3 + (b^2*c*d*x + b^3*d)*e^2 - (3*b*c^2*d^2*x + 2*b^2*c*d^2)*e)*sq
rt(c*x^2 + b*x))/(b^2*c^3*d^4*x^2 + b^3*c^2*d^4*x + (b^4*c*d^2*x^2 + b^5*d^2*x)*e^2 - 2*(b^3*c^2*d^3*x^2 + b^4
*c*d^3*x)*e), 2*((b^2*c*x^2 + b^3*x)*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))*e^2 - (2*c^3*d^3*x + b*c^2*d^3 + (b^2*c*d*x + b^3*d)*e^2 - (3*b*c^2*d^2*x + 2*b^2*c*d^2)*e)*sqrt(c
*x^2 + b*x))/(b^2*c^3*d^4*x^2 + b^3*c^2*d^4*x + (b^4*c*d^2*x^2 + b^5*d^2*x)*e^2 - 2*(b^3*c^2*d^3*x^2 + b^4*c*d
^3*x)*e)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.14, size = 167, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, c^{2} d^{2} - b c d e\right )} x}{b^{2} c d^{3} - b^{3} d^{2} e} + \frac {b c d^{2} - b^{2} d e}{b^{2} c d^{3} - b^{3} d^{2} e}\right )}}{\sqrt {c x^{2} + b x}} - \frac {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 ) e^{2}}{{\left (c d^{2} - b d e\right )} \sqrt {-c d^{2} + b d e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*c^2*d^2 - b*c*d*e)*x/(b^2*c*d^3 - b^3*d^2*e) + (b*c*d^2 - b^2*d*e)/(b^2*c*d^3 - b^3*d^2*e))/sqrt(c*x^2
+ b*x) - 2*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^2/((c*d^2 - b*d*e)*s
qrt(-c*d^2 + b*d*e))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________