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

Optimal. Leaf size=118 \[ \frac{c^2 \left (4 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^5}+\frac{x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+3 b c^2\right )}{8 d^4}+\frac{b x^3 \sqrt{d x-c} \sqrt{c+d x}}{4 d^2} \]

[Out]

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

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Rubi [A]  time = 0.317815, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{c^2 \left (4 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{4 d^5}+\frac{x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+3 b c^2\right )}{8 d^4}+\frac{b x^3 \sqrt{d x-c} \sqrt{c+d x}}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.9971, size = 104, normalized size = 0.88 \[ \frac{b x^{3} \sqrt{- c + d x} \sqrt{c + d x}}{4 d^{2}} + \frac{c^{2} \left (4 a d^{2} + 3 b c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d x}} \right )}}{4 d^{5}} + \frac{x \sqrt{- c + d x} \sqrt{c + d x} \left (4 a d^{2} + 3 b c^{2}\right )}{8 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**3*sqrt(-c + d*x)*sqrt(c + d*x)/(4*d**2) + c**2*(4*a*d**2 + 3*b*c**2)*atanh(
sqrt(c + d*x)/sqrt(-c + d*x))/(4*d**5) + x*sqrt(-c + d*x)*sqrt(c + d*x)*(4*a*d**
2 + 3*b*c**2)/(8*d**4)

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Mathematica [A]  time = 0.106383, size = 96, normalized size = 0.81 \[ \frac{d x \sqrt{d x-c} \sqrt{c+d x} \left (4 a d^2+3 b c^2+2 b d^2 x^2\right )+\left (4 a c^2 d^2+3 b c^4\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{8 d^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.03, size = 182, normalized size = 1.5 \[{\frac{{\it csgn} \left ( d \right ) }{8\,{d}^{5}}\sqrt{dx-c}\sqrt{dx+c} \left ( 2\,{\it csgn} \left ( d \right ){x}^{3}b{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+4\,ax\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) +3\,b{c}^{2}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d+4\,a{c}^{2}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){d}^{2}+3\,b{c}^{4}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(2*csgn(d)*x^3*b*d^3*(d^2*x^2-c^2)^(1/2)+4*a*x*(
d^2*x^2-c^2)^(1/2)*d^3*csgn(d)+3*b*c^2*x*(d^2*x^2-c^2)^(1/2)*csgn(d)*d+4*a*c^2*l
n((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))*d^2+3*b*c^4*ln((csgn(d)*(d^2*x^2-c^
2)^(1/2)+d*x)*csgn(d)))*csgn(d)/(d^2*x^2-c^2)^(1/2)/d^5

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Maxima [A]  time = 1.38641, size = 216, normalized size = 1.83 \[ \frac{\sqrt{d^{2} x^{2} - c^{2}} b x^{3}}{4 \, d^{2}} + \frac{3 \, b c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{8 \, \sqrt{d^{2}} d^{4}} + \frac{a c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} + \frac{3 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{2} x}{8 \, d^{4}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a x}{2 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(d^2*x^2 - c^2)*b*x^3/d^2 + 3/8*b*c^4*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2
)*sqrt(d^2))/(sqrt(d^2)*d^4) + 1/2*a*c^2*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*sqr
t(d^2))/(sqrt(d^2)*d^2) + 3/8*sqrt(d^2*x^2 - c^2)*b*c^2*x/d^4 + 1/2*sqrt(d^2*x^2
- c^2)*a*x/d^2

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Fricas [A]  time = 0.252791, size = 508, normalized size = 4.31 \[ -\frac{16 \, b d^{8} x^{8} + 32 \, a d^{8} x^{6} - 4 \,{\left (7 \, b c^{4} d^{4} + 12 \, a c^{2} d^{6}\right )} x^{4} + 4 \,{\left (3 \, b c^{6} d^{2} + 4 \, a c^{4} d^{4}\right )} x^{2} -{\left (16 \, b d^{7} x^{7} + 8 \,{\left (b c^{2} d^{5} + 4 \, a d^{7}\right )} x^{5} - 2 \,{\left (11 \, b c^{4} d^{3} + 16 \, a c^{2} d^{5}\right )} x^{3} +{\left (3 \, b c^{6} d + 4 \, a c^{4} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} +{\left (3 \, b c^{8} + 4 \, a c^{6} d^{2} + 8 \,{\left (3 \, b c^{4} d^{4} + 4 \, a c^{2} d^{6}\right )} x^{4} - 8 \,{\left (3 \, b c^{6} d^{2} + 4 \, a c^{4} d^{4}\right )} x^{2} - 4 \,{\left (2 \,{\left (3 \, b c^{4} d^{3} + 4 \, a c^{2} d^{5}\right )} x^{3} -{\left (3 \, b c^{6} d + 4 \, a c^{4} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{8 \,{\left (8 \, d^{9} x^{4} - 8 \, c^{2} d^{7} x^{2} + c^{4} d^{5} - 4 \,{\left (2 \, d^{8} x^{3} - c^{2} d^{6} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(16*b*d^8*x^8 + 32*a*d^8*x^6 - 4*(7*b*c^4*d^4 + 12*a*c^2*d^6)*x^4 + 4*(3*b*
c^6*d^2 + 4*a*c^4*d^4)*x^2 - (16*b*d^7*x^7 + 8*(b*c^2*d^5 + 4*a*d^7)*x^5 - 2*(11
*b*c^4*d^3 + 16*a*c^2*d^5)*x^3 + (3*b*c^6*d + 4*a*c^4*d^3)*x)*sqrt(d*x + c)*sqrt
(d*x - c) + (3*b*c^8 + 4*a*c^6*d^2 + 8*(3*b*c^4*d^4 + 4*a*c^2*d^6)*x^4 - 8*(3*b*
c^6*d^2 + 4*a*c^4*d^4)*x^2 - 4*(2*(3*b*c^4*d^3 + 4*a*c^2*d^5)*x^3 - (3*b*c^6*d +
4*a*c^4*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))*log(-d*x + sqrt(d*x + c)*sqrt(d*x
- c)))/(8*d^9*x^4 - 8*c^2*d^7*x^2 + c^4*d^5 - 4*(2*d^8*x^3 - c^2*d^6*x)*sqrt(d*x
+ c)*sqrt(d*x - c))

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Sympy [A]  time = 117.765, size = 236, normalized size = 2. \[ \frac{a c^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i a c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{b c^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{5}} - \frac{i b c^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**2*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0)
, ()), c**2/(d**2*x**2))/(4*pi**(3/2)*d**3) - I*a*c**2*meijerg(((-3/2, -5/4, -1,
-3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)), c**2*exp_polar(2*I*pi)/
(d**2*x**2))/(4*pi**(3/2)*d**3) + b*c**4*meijerg(((-7/4, -5/4), (-3/2, -3/2, -1,
1)), ((-2, -7/4, -3/2, -5/4, -1, 0), ()), c**2/(d**2*x**2))/(4*pi**(3/2)*d**5)
- I*b*c**4*meijerg(((-5/2, -9/4, -2, -7/4, -3/2, 1), ()), ((-9/4, -7/4), (-5/2,
-2, -2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**5)

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GIAC/XCAS [A]  time = 0.255064, size = 176, normalized size = 1.49 \[ -\frac{{\left (5 \, b c^{3} d^{16} + 4 \, a c d^{18} -{\left (9 \, b c^{2} d^{16} + 4 \, a d^{18} + 2 \,{\left ({\left (d x + c\right )} b d^{16} - 3 \, b c d^{16}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c} + 2 \,{\left (3 \, b c^{4} d^{16} + 4 \, a c^{2} d^{18}\right )}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{114688 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/114688*((5*b*c^3*d^16 + 4*a*c*d^18 - (9*b*c^2*d^16 + 4*a*d^18 + 2*((d*x + c)*
b*d^16 - 3*b*c*d^16)*(d*x + c))*(d*x + c))*sqrt(d*x + c)*sqrt(d*x - c) + 2*(3*b*
c^4*d^16 + 4*a*c^2*d^18)*ln(abs(-sqrt(d*x + c) + sqrt(d*x - c))))/d