3.480 \(\int \frac{x}{\left (a+b \sqrt{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt{c+d x}\right )}+\frac{2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{4 a \sqrt{c+d x}}{b^3 d^2}+\frac{x}{b^2 d} \]

[Out]

x/(b^2*d) - (4*a*Sqrt[c + d*x])/(b^3*d^2) + (2*a*(a^2 - b^2*c))/(b^4*d^2*(a + b*
Sqrt[c + d*x])) + (2*(3*a^2 - b^2*c)*Log[a + b*Sqrt[c + d*x]])/(b^4*d^2)

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Rubi [A]  time = 0.199244, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2 a \left (a^2-b^2 c\right )}{b^4 d^2 \left (a+b \sqrt{c+d x}\right )}+\frac{2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{4 a \sqrt{c+d x}}{b^3 d^2}+\frac{x}{b^2 d} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[c + d*x])^2,x]

[Out]

x/(b^2*d) - (4*a*Sqrt[c + d*x])/(b^3*d^2) + (2*a*(a^2 - b^2*c))/(b^4*d^2*(a + b*
Sqrt[c + d*x])) + (2*(3*a^2 - b^2*c)*Log[a + b*Sqrt[c + d*x]])/(b^4*d^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 a \sqrt{c + d x}}{b^{3} d^{2}} + \frac{2 a \left (a^{2} - b^{2} c\right )}{b^{4} d^{2} \left (a + b \sqrt{c + d x}\right )} + \frac{2 \int ^{\sqrt{c + d x}} x\, dx}{b^{2} d^{2}} + \frac{2 \left (3 a^{2} - b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*(d*x+c)**(1/2))**2,x)

[Out]

-4*a*sqrt(c + d*x)/(b**3*d**2) + 2*a*(a**2 - b**2*c)/(b**4*d**2*(a + b*sqrt(c +
d*x))) + 2*Integral(x, (x, sqrt(c + d*x)))/(b**2*d**2) + 2*(3*a**2 - b**2*c)*log
(a + b*sqrt(c + d*x))/(b**4*d**2)

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Mathematica [A]  time = 0.123418, size = 86, normalized size = 0.91 \[ \frac{\frac{2 \left (a^3-a b^2 c\right )}{a+b \sqrt{c+d x}}+2 \left (3 a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )-4 a b \sqrt{c+d x}+b^2 (c+d x)}{b^4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a*b*Sqrt[c + d*x] + b^2*(c + d*x) + (2*(a^3 - a*b^2*c))/(a + b*Sqrt[c + d*x]
) + 2*(3*a^2 - b^2*c)*Log[a + b*Sqrt[c + d*x]])/(b^4*d^2)

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Maple [A]  time = 0.012, size = 125, normalized size = 1.3 \[{\frac{x}{{b}^{2}d}}+{\frac{c}{{b}^{2}{d}^{2}}}-4\,{\frac{a\sqrt{dx+c}}{{b}^{3}{d}^{2}}}-2\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{2}{d}^{2}}}+6\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{2}}{{b}^{4}{d}^{2}}}-2\,{\frac{ac}{{b}^{2}{d}^{2} \left ( a+b\sqrt{dx+c} \right ) }}+2\,{\frac{{a}^{3}}{{b}^{4}{d}^{2} \left ( a+b\sqrt{dx+c} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*(d*x+c)^(1/2))^2,x)

[Out]

x/b^2/d+1/d^2/b^2*c-4*a*(d*x+c)^(1/2)/b^3/d^2-2/d^2/b^2*ln(a+b*(d*x+c)^(1/2))*c+
6/d^2/b^4*ln(a+b*(d*x+c)^(1/2))*a^2-2/d^2*a/b^2/(a+b*(d*x+c)^(1/2))*c+2/d^2*a^3/
b^4/(a+b*(d*x+c)^(1/2))

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Maxima [A]  time = 0.693321, size = 122, normalized size = 1.28 \[ -\frac{\frac{2 \,{\left (a b^{2} c - a^{3}\right )}}{\sqrt{d x + c} b^{5} + a b^{4}} - \frac{{\left (d x + c\right )} b - 4 \, \sqrt{d x + c} a}{b^{3}} + \frac{2 \,{\left (b^{2} c - 3 \, a^{2}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{4}}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(d*x + c)*b + a)^2,x, algorithm="maxima")

[Out]

-(2*(a*b^2*c - a^3)/(sqrt(d*x + c)*b^5 + a*b^4) - ((d*x + c)*b - 4*sqrt(d*x + c)
*a)/b^3 + 2*(b^2*c - 3*a^2)*log(sqrt(d*x + c)*b + a)/b^4)/d^2

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Fricas [A]  time = 0.266919, size = 163, normalized size = 1.72 \[ -\frac{3 \, a b^{2} d x + 5 \, a b^{2} c - 2 \, a^{3} + 2 \,{\left (a b^{2} c - 3 \, a^{3} +{\left (b^{3} c - 3 \, a^{2} b\right )} \sqrt{d x + c}\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (b^{3} d x + b^{3} c - 4 \, a^{2} b\right )} \sqrt{d x + c}}{\sqrt{d x + c} b^{5} d^{2} + a b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(d*x + c)*b + a)^2,x, algorithm="fricas")

[Out]

-(3*a*b^2*d*x + 5*a*b^2*c - 2*a^3 + 2*(a*b^2*c - 3*a^3 + (b^3*c - 3*a^2*b)*sqrt(
d*x + c))*log(sqrt(d*x + c)*b + a) - (b^3*d*x + b^3*c - 4*a^2*b)*sqrt(d*x + c))/
(sqrt(d*x + c)*b^5*d^2 + a*b^4*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b \sqrt{c + d x}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*(d*x+c)**(1/2))**2,x)

[Out]

Integral(x/(a + b*sqrt(c + d*x))**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(d*x + c)*b + a)^2,x, algorithm="giac")

[Out]

undef