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

Optimal. Leaf size=151 \[ -\frac{2 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{2 \left (a^2-b^2 c\right )^2 \sqrt{c+d x}}{b^5 d^3}-\frac{a x \left (a^2-2 b^2 c\right )}{b^4 d^2}+\frac{2 \left (a^2-2 b^2 c\right ) (c+d x)^{3/2}}{3 b^3 d^3}-\frac{a (c+d x)^2}{2 b^2 d^3}+\frac{2 (c+d x)^{5/2}}{5 b d^3} \]

[Out]

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

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Rubi [A]  time = 0.327907, antiderivative size = 151, 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 \[ -\frac{2 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{2 \left (a^2-b^2 c\right )^2 \sqrt{c+d x}}{b^5 d^3}-\frac{a x \left (a^2-2 b^2 c\right )}{b^4 d^2}+\frac{2 \left (a^2-2 b^2 c\right ) (c+d x)^{3/2}}{3 b^3 d^3}-\frac{a (c+d x)^2}{2 b^2 d^3}+\frac{2 (c+d x)^{5/2}}{5 b d^3} \]

Antiderivative was successfully verified.

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

[Out]

-((a*(a^2 - 2*b^2*c)*x)/(b^4*d^2)) + (2*(a^2 - b^2*c)^2*Sqrt[c + d*x])/(b^5*d^3)
+ (2*(a^2 - 2*b^2*c)*(c + d*x)^(3/2))/(3*b^3*d^3) - (a*(c + d*x)^2)/(2*b^2*d^3)
+ (2*(c + d*x)^(5/2))/(5*b*d^3) - (2*a*(a^2 - b^2*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. \[ - \frac{a \left (c + d x\right )^{2}}{2 b^{2} d^{3}} - \frac{2 a \left (a^{2} - 2 b^{2} c\right ) \int ^{\sqrt{c + d x}} x\, dx}{b^{4} d^{3}} - \frac{2 a \left (a^{2} - b^{2} c\right )^{2} \log{\left (a + b \sqrt{c + d x} \right )}}{b^{6} d^{3}} + \frac{2 \left (a^{2} - b^{2} c\right )^{2} \int ^{\sqrt{c + d x}} \frac{1}{b^{5}}\, dx}{d^{3}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 b d^{3}} + \frac{2 \left (a^{2} - 2 b^{2} c\right ) \left (c + d x\right )^{\frac{3}{2}}}{3 b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.351033, size = 169, normalized size = 1.12 \[ \frac{-30 a \left (a^2-b^2 c\right )^2 \log \left (a^2-b^2 (c+d x)\right )-60 a \left (a^2-b^2 c\right )^2 \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )+b \left (60 a^4 \sqrt{c+d x}-30 a^3 b d x-20 a^2 b^2 (5 c-d x) \sqrt{c+d x}-15 a b^3 d x (d x-2 c)+4 b^4 \sqrt{c+d x} \left (8 c^2-4 c d x+3 d^2 x^2\right )\right )}{30 b^6 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b*(-30*a^3*b*d*x - 15*a*b^3*d*x*(-2*c + d*x) + 60*a^4*Sqrt[c + d*x] - 20*a^2*b^
2*(5*c - d*x)*Sqrt[c + d*x] + 4*b^4*Sqrt[c + d*x]*(8*c^2 - 4*c*d*x + 3*d^2*x^2))
- 60*a*(a^2 - b^2*c)^2*ArcTanh[(b*Sqrt[c + d*x])/a] - 30*a*(a^2 - b^2*c)^2*Log[
a^2 - b^2*(c + d*x)])/(30*b^6*d^3)

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Maple [A]  time = 0.007, size = 235, normalized size = 1.6 \[{\frac{2}{5\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{a{x}^{2}}{2\,{b}^{2}d}}+{\frac{acx}{{b}^{2}{d}^{2}}}+{\frac{3\,a{c}^{2}}{2\,{b}^{2}{d}^{3}}}-{\frac{4\,c}{3\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{a}^{2}}{3\,{b}^{3}{d}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{c}^{2}\sqrt{dx+c}}{b{d}^{3}}}-{\frac{{a}^{3}x}{{b}^{4}{d}^{2}}}-{\frac{{a}^{3}c}{{b}^{4}{d}^{3}}}-4\,{\frac{{a}^{2}c\sqrt{dx+c}}{{b}^{3}{d}^{3}}}+2\,{\frac{\sqrt{dx+c}{a}^{4}}{{d}^{3}{b}^{5}}}-2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ){c}^{2}}{{b}^{2}{d}^{3}}}+4\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{4}{d}^{3}}}-2\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{dx+c} \right ) }{{d}^{3}{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*(d*x+c)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.704924, size = 200, normalized size = 1.32 \[ \frac{\frac{12 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} - 15 \,{\left (d x + c\right )}^{2} a b^{3} - 20 \,{\left (2 \, b^{4} c - a^{2} b^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 30 \,{\left (2 \, a b^{3} c - a^{3} b\right )}{\left (d x + c\right )} + 60 \,{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{d x + c}}{b^{5}} - \frac{60 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{6}}}{30 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*((12*(d*x + c)^(5/2)*b^4 - 15*(d*x + c)^2*a*b^3 - 20*(2*b^4*c - a^2*b^2)*(d
*x + c)^(3/2) + 30*(2*a*b^3*c - a^3*b)*(d*x + c) + 60*(b^4*c^2 - 2*a^2*b^2*c + a
^4)*sqrt(d*x + c))/b^5 - 60*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*log(sqrt(d*x + c)*b
+ a)/b^6)/d^3

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Fricas [A]  time = 0.293567, size = 186, normalized size = 1.23 \[ -\frac{15 \, a b^{4} d^{2} x^{2} - 30 \,{\left (a b^{4} c - a^{3} b^{2}\right )} d x + 60 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 4 \,{\left (3 \, b^{5} d^{2} x^{2} + 8 \, b^{5} c^{2} - 25 \, a^{2} b^{3} c + 15 \, a^{4} b -{\left (4 \, b^{5} c - 5 \, a^{2} b^{3}\right )} d x\right )} \sqrt{d x + c}}{30 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(15*a*b^4*d^2*x^2 - 30*(a*b^4*c - a^3*b^2)*d*x + 60*(a*b^4*c^2 - 2*a^3*b^2
*c + a^5)*log(sqrt(d*x + c)*b + a) - 4*(3*b^5*d^2*x^2 + 8*b^5*c^2 - 25*a^2*b^3*c
+ 15*a^4*b - (4*b^5*c - 5*a^2*b^3)*d*x)*sqrt(d*x + c))/(b^6*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283787, size = 320, normalized size = 2.12 \[ -\frac{2 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )}{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{6} d^{3}} + \frac{2 \,{\left (a b^{4} c^{2}{\rm ln}\left ({\left | a \right |}\right ) - 2 \, a^{3} b^{2} c{\rm ln}\left ({\left | a \right |}\right ) + a^{5}{\rm ln}\left ({\left | a \right |}\right )\right )}}{b^{6} d^{3}} + \frac{12 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} d^{12} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c d^{12} + 60 \, \sqrt{d x + c} b^{4} c^{2} d^{12} - 15 \,{\left (d x + c\right )}^{2} a b^{3} d^{12} + 60 \,{\left (d x + c\right )} a b^{3} c d^{12} + 20 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} d^{12} - 120 \, \sqrt{d x + c} a^{2} b^{2} c d^{12} - 30 \,{\left (d x + c\right )} a^{3} b d^{12} + 60 \, \sqrt{d x + c} a^{4} d^{12}}{30 \, b^{5} d^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*ln(abs(sqrt(d*x + c)*b + a))/(b^6*d^3) + 2*(a
*b^4*c^2*ln(abs(a)) - 2*a^3*b^2*c*ln(abs(a)) + a^5*ln(abs(a)))/(b^6*d^3) + 1/30*
(12*(d*x + c)^(5/2)*b^4*d^12 - 40*(d*x + c)^(3/2)*b^4*c*d^12 + 60*sqrt(d*x + c)*
b^4*c^2*d^12 - 15*(d*x + c)^2*a*b^3*d^12 + 60*(d*x + c)*a*b^3*c*d^12 + 20*(d*x +
c)^(3/2)*a^2*b^2*d^12 - 120*sqrt(d*x + c)*a^2*b^2*c*d^12 - 30*(d*x + c)*a^3*b*d
^12 + 60*sqrt(d*x + c)*a^4*d^12)/(b^5*d^15)