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

Optimal. Leaf size=166 \[ \frac{2 a \left (a^2-b^2 c\right )^2}{b^6 d^3 \left (a+b \sqrt{c+d x}\right )}-\frac{8 a \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^5 d^3}+\frac{x \left (3 a^2-2 b^2 c\right )}{b^4 d^2}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}-\frac{4 a (c+d x)^{3/2}}{3 b^3 d^3}+\frac{(c+d x)^2}{2 b^2 d^3} \]

[Out]

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

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Rubi [A]  time = 0.371511, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 a \left (a^2-b^2 c\right )^2}{b^6 d^3 \left (a+b \sqrt{c+d x}\right )}-\frac{8 a \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^5 d^3}+\frac{x \left (3 a^2-2 b^2 c\right )}{b^4 d^2}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}-\frac{4 a (c+d x)^{3/2}}{3 b^3 d^3}+\frac{(c+d x)^2}{2 b^2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 a \left (c + d x\right )^{\frac{3}{2}}}{3 b^{3} d^{3}} - \frac{8 a \left (a^{2} - b^{2} c\right ) \sqrt{c + d x}}{b^{5} d^{3}} + \frac{2 a \left (a^{2} - b^{2} c\right )^{2}}{b^{6} d^{3} \left (a + b \sqrt{c + d x}\right )} + \frac{\left (c + d x\right )^{2}}{2 b^{2} d^{3}} + \frac{2 \left (3 a^{2} - 2 b^{2} c\right ) \int ^{\sqrt{c + d x}} x\, dx}{b^{4} d^{3}} + \frac{2 \left (a^{2} - b^{2} c\right ) \left (5 a^{2} - b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a*(c + d*x)**(3/2)/(3*b**3*d**3) - 8*a*(a**2 - b**2*c)*sqrt(c + d*x)/(b**5*d*
*3) + 2*a*(a**2 - b**2*c)**2/(b**6*d**3*(a + b*sqrt(c + d*x))) + (c + d*x)**2/(2
*b**2*d**3) + 2*(3*a**2 - 2*b**2*c)*Integral(x, (x, sqrt(c + d*x)))/(b**4*d**3)
+ 2*(a**2 - b**2*c)*(5*a**2 - b**2*c)*log(a + b*sqrt(c + d*x))/(b**6*d**3)

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Mathematica [A]  time = 0.357992, size = 224, normalized size = 1.35 \[ \frac{-6 b^2 d x \left (b^2 c-3 a^2\right )+12 \left (a^2-b^2 c\right ) \left (5 a^2-b^2 c\right ) \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )+\frac{4 a b \sqrt{c+d x} \left (15 a^4-2 a^2 b^2 (14 c+5 d x)+b^4 \left (13 c^2+8 c d x-2 d^2 x^2\right )\right )}{b^2 (c+d x)-a^2}+6 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \log \left (a^2-b^2 (c+d x)\right )+\frac{12 \left (a^3-a b^2 c\right )^2}{a^2-b^2 (c+d x)}+3 b^4 d^2 x^2}{6 b^6 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-6*b^2*(-3*a^2 + b^2*c)*d*x + 3*b^4*d^2*x^2 + (12*(a^3 - a*b^2*c)^2)/(a^2 - b^2
*(c + d*x)) + (4*a*b*Sqrt[c + d*x]*(15*a^4 - 2*a^2*b^2*(14*c + 5*d*x) + b^4*(13*
c^2 + 8*c*d*x - 2*d^2*x^2)))/(-a^2 + b^2*(c + d*x)) + 12*(a^2 - b^2*c)*(5*a^2 -
b^2*c)*ArcTanh[(b*Sqrt[c + d*x])/a] + 6*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*Log[a^2
- b^2*(c + d*x)])/(6*b^6*d^3)

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Maple [A]  time = 0.013, size = 253, normalized size = 1.5 \[{\frac{{x}^{2}}{2\,{b}^{2}d}}-{\frac{cx}{{b}^{2}{d}^{2}}}-{\frac{3\,{c}^{2}}{2\,{b}^{2}{d}^{3}}}-{\frac{4\,a}{3\,{b}^{3}{d}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+3\,{\frac{{a}^{2}x}{{b}^{4}{d}^{2}}}+3\,{\frac{{a}^{2}c}{{b}^{4}{d}^{3}}}+8\,{\frac{ac\sqrt{dx+c}}{{b}^{3}{d}^{3}}}-8\,{\frac{{a}^{3}\sqrt{dx+c}}{{d}^{3}{b}^{5}}}+2\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){c}^{2}}{{b}^{2}{d}^{3}}}-12\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ) c{a}^{2}}{{b}^{4}{d}^{3}}}+10\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{4}}{{d}^{3}{b}^{6}}}+2\,{\frac{a{c}^{2}}{{b}^{2}{d}^{3} \left ( a+b\sqrt{dx+c} \right ) }}-4\,{\frac{{a}^{3}c}{{b}^{4}{d}^{3} \left ( a+b\sqrt{dx+c} \right ) }}+2\,{\frac{{a}^{5}}{{d}^{3}{b}^{6} \left ( a+b\sqrt{dx+c} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/d/b^2*x^2-1/d^2/b^2*x*c-3/2/d^3/b^2*c^2-4/3*a*(d*x+c)^(3/2)/b^3/d^3+3/d^2/b^
4*x*a^2+3/d^3/b^4*a^2*c+8/d^3/b^3*a*c*(d*x+c)^(1/2)-8/d^3/b^5*a^3*(d*x+c)^(1/2)+
2/d^3/b^2*ln(a+b*(d*x+c)^(1/2))*c^2-12/d^3/b^4*ln(a+b*(d*x+c)^(1/2))*c*a^2+10/d^
3/b^6*ln(a+b*(d*x+c)^(1/2))*a^4+2/d^3*a/b^2/(a+b*(d*x+c)^(1/2))*c^2-4/d^3*a^3/b^
4/(a+b*(d*x+c)^(1/2))*c+2/d^3*a^5/b^6/(a+b*(d*x+c)^(1/2))

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Maxima [A]  time = 0.703476, size = 213, normalized size = 1.28 \[ \frac{\frac{12 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )}}{\sqrt{d x + c} b^{7} + a b^{6}} + \frac{3 \,{\left (d x + c\right )}^{2} b^{3} - 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} - 6 \,{\left (2 \, b^{3} c - 3 \, a^{2} b\right )}{\left (d x + c\right )} + 48 \,{\left (a b^{2} c - a^{3}\right )} \sqrt{d x + c}}{b^{5}} + \frac{12 \,{\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{6}}}{6 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(12*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)/(sqrt(d*x + c)*b^7 + a*b^6) + (3*(d*x +
c)^2*b^3 - 8*(d*x + c)^(3/2)*a*b^2 - 6*(2*b^3*c - 3*a^2*b)*(d*x + c) + 48*(a*b^2
*c - a^3)*sqrt(d*x + c))/b^5 + 12*(b^4*c^2 - 6*a^2*b^2*c + 5*a^4)*log(sqrt(d*x +
c)*b + a)/b^6)/d^3

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Fricas [A]  time = 0.268676, size = 289, normalized size = 1.74 \[ -\frac{5 \, a b^{4} d^{2} x^{2} - 43 \, a b^{4} c^{2} + 54 \, a^{3} b^{2} c - 12 \, a^{5} - 2 \,{\left (13 \, a b^{4} c - 15 \, a^{3} b^{2}\right )} d x - 12 \,{\left (a b^{4} c^{2} - 6 \, a^{3} b^{2} c + 5 \, a^{5} +{\left (b^{5} c^{2} - 6 \, a^{2} b^{3} c + 5 \, a^{4} b\right )} \sqrt{d x + c}\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (3 \, b^{5} d^{2} x^{2} - 9 \, b^{5} c^{2} + 58 \, a^{2} b^{3} c - 48 \, a^{4} b - 2 \,{\left (3 \, b^{5} c - 5 \, a^{2} b^{3}\right )} d x\right )} \sqrt{d x + c}}{6 \,{\left (\sqrt{d x + c} b^{7} d^{3} + a b^{6} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(5*a*b^4*d^2*x^2 - 43*a*b^4*c^2 + 54*a^3*b^2*c - 12*a^5 - 2*(13*a*b^4*c - 1
5*a^3*b^2)*d*x - 12*(a*b^4*c^2 - 6*a^3*b^2*c + 5*a^5 + (b^5*c^2 - 6*a^2*b^3*c +
5*a^4*b)*sqrt(d*x + c))*log(sqrt(d*x + c)*b + a) - (3*b^5*d^2*x^2 - 9*b^5*c^2 +
58*a^2*b^3*c - 48*a^4*b - 2*(3*b^5*c - 5*a^2*b^3)*d*x)*sqrt(d*x + c))/(sqrt(d*x
+ c)*b^7*d^3 + a*b^6*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(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^2/(sqrt(d*x + c)*b + a)^2,x, algorithm="giac")

[Out]

undef