3.4.24 \(\int \frac {d+e x}{(b x+c x^2)^{3/2}} \, dx\) [324]

Optimal. Leaf size=33 \[ -\frac {2 (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {650} \begin {gather*} -\frac {2 (x (2 c d-b e)+b d)}{b^2 \sqrt {b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(b*x + c*x^2)^(3/2),x]

[Out]

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

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 30, normalized size = 0.91 \begin {gather*} \frac {2 (-b d-2 c d x+b e x)}{b^2 \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(31)=62\).
time = 0.47, size = 68, normalized size = 2.06

method	result	size
gosper	\(-\frac {2 x \left (c x +b \right ) \left (-b e x +2 c d x +b d \right )}{b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\)	\(37\)
trager	\(-\frac {2 \left (-b e x +2 c d x +b d \right ) \sqrt {c \,x^{2}+b x}}{\left (c x +b \right ) b^{2} x}\)	\(41\)
risch	\(-\frac {2 d \left (c x +b \right )}{b^{2} \sqrt {x \left (c x +b \right )}}+\frac {2 \left (b e -c d \right ) x}{\sqrt {x \left (c x +b \right )}\, b^{2}}\)	\(45\)
default	\(e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )-\frac {2 d \left (2 c x +b \right )}{b^{2} \sqrt {c \,x^{2}+b x}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 56, normalized size = 1.70 \begin {gather*} -\frac {4 \, c d x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, x e}{\sqrt {c x^{2} + b x} b} - \frac {2 \, d}{\sqrt {c x^{2} + b x} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c*d*x/(sqrt(c*x^2 + b*x)*b^2) + 2*x*e/(sqrt(c*x^2 + b*x)*b) - 2*d/(sqrt(c*x^2 + b*x)*b)

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Fricas [A]
time = 1.53, size = 44, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (2 \, c d x - b x e + b d\right )} \sqrt {c x^{2} + b x}}{b^{2} c x^{2} + b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.44, size = 34, normalized size = 1.03 \begin {gather*} -\frac {2 \, {\left (\frac {d}{b} + \frac {{\left (2 \, c d - b e\right )} x}{b^{2}}\right )}}{\sqrt {c x^{2} + b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.27, size = 31, normalized size = 0.94 \begin {gather*} -\frac {2\,b\,d-2\,b\,e\,x+4\,c\,d\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(b*x + c*x^2)^(3/2),x)

[Out]

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

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