3.473 \(\int \frac{x}{a+b \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.178688, antiderivative size = 90, 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 \[ -\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}+\frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[c + d*x]),x]

[Out]

-((a*x)/(b^2*d)) + (2*(a^2 - b^2*c)*Sqrt[c + d*x])/(b^3*d^2) + (2*(c + d*x)^(3/2
))/(3*b*d^2) - (2*a*(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. \[ - \frac{2 a \int ^{\sqrt{c + d x}} x\, dx}{b^{2} d^{2}} - \frac{2 a \left (a^{2} - b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} + \frac{2 \left (a^{2} - b^{2} c\right ) \int ^{\sqrt{c + d x}} \frac{1}{b^{3}}\, dx}{d^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.083584, size = 85, normalized size = 0.94 \[ \frac{b \left (6 a^2 \sqrt{c+d x}-3 a b (c+d x)+2 b^2 (d x-2 c) \sqrt{c+d x}\right )-6 \left (a^3-a b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{3 b^4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(b*(6*a^2*Sqrt[c + d*x] + 2*b^2*(-2*c + d*x)*Sqrt[c + d*x] - 3*a*b*(c + d*x)) -
6*(a^3 - a*b^2*c)*Log[a + b*Sqrt[c + d*x]])/(3*b^4*d^2)

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Maple [A]  time = 0.006, size = 116, normalized size = 1.3 \[{\frac{2}{3\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{{b}^{2}d}}-{\frac{ac}{{b}^{2}{d}^{2}}}-2\,{\frac{c\sqrt{dx+c}}{b{d}^{2}}}+2\,{\frac{\sqrt{dx+c}{a}^{2}}{{b}^{3}{d}^{2}}}+2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{2}{d}^{2}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ) }{{b}^{4}{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*(d*x+c)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.699251, size = 109, normalized size = 1.21 \[ \frac{\frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} - 3 \,{\left (d x + c\right )} a b - 6 \,{\left (b^{2} c - a^{2}\right )} \sqrt{d x + c}}{b^{3}} + \frac{6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{4}}}{3 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28963, size = 96, normalized size = 1.07 \[ -\frac{3 \, a b^{2} d x - 6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 2 \,{\left (b^{3} d x - 2 \, b^{3} c + 3 \, a^{2} b\right )} \sqrt{d x + c}}{3 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(a + b*sqrt(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.276789, size = 177, normalized size = 1.97 \[ \frac{\frac{6 \,{\left (a b^{2} c - a^{3}\right )}{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{4} d} - \frac{6 \,{\left (a b^{2} c{\rm ln}\left ({\left | a \right |}\right ) - a^{3}{\rm ln}\left ({\left | a \right |}\right )\right )}}{b^{4} d} + \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{2} - 6 \, \sqrt{d x + c} b^{2} c d^{2} - 3 \,{\left (d x + c\right )} a b d^{2} + 6 \, \sqrt{d x + c} a^{2} d^{2}}{b^{3} d^{3}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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