3.244 \(\int x^4 \sqrt{-c+d x} \sqrt{c+d x} \left (a+b x^2\right ) \, dx\)

Optimal. Leaf size=208 \[ \frac{c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}+\frac{x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}-\frac{c^6 \left (8 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{64 d^7}+\frac{c^4 x \sqrt{d x-c} \sqrt{c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac{b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]

[Out]

(c^4*(5*b*c^2 + 8*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(128*d^6) + (c^2*(5*b*c
^2 + 8*a*d^2)*x*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(64*d^6) + ((5*b*c^2 + 8*a*d^2
)*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(48*d^4) + (b*x^5*(-c + d*x)^(3/2)*(c +
d*x)^(3/2))/(8*d^2) - (c^6*(5*b*c^2 + 8*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d
*x]])/(64*d^7)

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Rubi [A]  time = 0.476242, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{c^2 x (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{64 d^6}+\frac{x^3 (d x-c)^{3/2} (c+d x)^{3/2} \left (8 a d^2+5 b c^2\right )}{48 d^4}-\frac{c^6 \left (8 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{64 d^7}+\frac{c^4 x \sqrt{d x-c} \sqrt{c+d x} \left (8 a d^2+5 b c^2\right )}{128 d^6}+\frac{b x^5 (d x-c)^{3/2} (c+d x)^{3/2}}{8 d^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

(c^4*(5*b*c^2 + 8*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(128*d^6) + (c^2*(5*b*c
^2 + 8*a*d^2)*x*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(64*d^6) + ((5*b*c^2 + 8*a*d^2
)*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(48*d^4) + (b*x^5*(-c + d*x)^(3/2)*(c +
d*x)^(3/2))/(8*d^2) - (c^6*(5*b*c^2 + 8*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d
*x]])/(64*d^7)

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Rubi in Sympy [A]  time = 34.3333, size = 187, normalized size = 0.9 \[ \frac{b x^{5} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{8 d^{2}} - \frac{c^{6} \left (8 a d^{2} + 5 b c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d x}} \right )}}{64 d^{7}} + \frac{c^{4} x \sqrt{- c + d x} \sqrt{c + d x} \left (8 a d^{2} + 5 b c^{2}\right )}{128 d^{6}} + \frac{c^{2} x \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (8 a d^{2} + 5 b c^{2}\right )}{64 d^{6}} + \frac{x^{3} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (8 a d^{2} + 5 b c^{2}\right )}{48 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2),x)

[Out]

b*x**5*(-c + d*x)**(3/2)*(c + d*x)**(3/2)/(8*d**2) - c**6*(8*a*d**2 + 5*b*c**2)*
atanh(sqrt(c + d*x)/sqrt(-c + d*x))/(64*d**7) + c**4*x*sqrt(-c + d*x)*sqrt(c + d
*x)*(8*a*d**2 + 5*b*c**2)/(128*d**6) + c**2*x*(-c + d*x)**(3/2)*(c + d*x)**(3/2)
*(8*a*d**2 + 5*b*c**2)/(64*d**6) + x**3*(-c + d*x)**(3/2)*(c + d*x)**(3/2)*(8*a*
d**2 + 5*b*c**2)/(48*d**4)

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Mathematica [A]  time = 0.177939, size = 146, normalized size = 0.7 \[ \frac{d x \sqrt{d x-c} \sqrt{c+d x} \left (8 a d^2 \left (-3 c^4-2 c^2 d^2 x^2+8 d^4 x^4\right )-b \left (15 c^6+10 c^4 d^2 x^2+8 c^2 d^4 x^4-48 d^6 x^6\right )\right )-3 \left (8 a c^6 d^2+5 b c^8\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{384 d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

(d*x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(8*a*d^2*(-3*c^4 - 2*c^2*d^2*x^2 + 8*d^4*x^4)
- b*(15*c^6 + 10*c^4*d^2*x^2 + 8*c^2*d^4*x^4 - 48*d^6*x^6)) - 3*(5*b*c^8 + 8*a*c
^6*d^2)*Log[d*x + Sqrt[-c + d*x]*Sqrt[c + d*x]])/(384*d^7)

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Maple [C]  time = 0.032, size = 298, normalized size = 1.4 \[{\frac{{\it csgn} \left ( d \right ) }{384\,{d}^{7}}\sqrt{dx-c}\sqrt{dx+c} \left ( 48\,{\it csgn} \left ( d \right ){x}^{7}b{d}^{7}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+64\,{\it csgn} \left ( d \right ){x}^{5}a{d}^{7}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-8\,{\it csgn} \left ( d \right ){x}^{5}b{c}^{2}{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-16\,{\it csgn} \left ( d \right ){x}^{3}a{c}^{2}{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-10\,{\it csgn} \left ( d \right ){x}^{3}b{c}^{4}{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}-24\,a{c}^{4}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) -15\,b{c}^{6}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d-24\,a{c}^{6}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){d}^{2}-15\,b{c}^{8}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2),x)

[Out]

1/384*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(48*csgn(d)*x^7*b*d^7*(d^2*x^2-c^2)^(1/2)+64*c
sgn(d)*x^5*a*d^7*(d^2*x^2-c^2)^(1/2)-8*csgn(d)*x^5*b*c^2*d^5*(d^2*x^2-c^2)^(1/2)
-16*csgn(d)*x^3*a*c^2*d^5*(d^2*x^2-c^2)^(1/2)-10*csgn(d)*x^3*b*c^4*d^3*(d^2*x^2-
c^2)^(1/2)-24*a*c^4*x*(d^2*x^2-c^2)^(1/2)*d^3*csgn(d)-15*b*c^6*x*(d^2*x^2-c^2)^(
1/2)*csgn(d)*d-24*a*c^6*ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))*d^2-15*b*c
^8*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^
7

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Maxima [A]  time = 1.41082, size = 356, normalized size = 1.71 \[ \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x^{5}}{8 \, d^{2}} + \frac{5 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{2} x^{3}}{48 \, d^{4}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a x^{3}}{6 \, d^{2}} - \frac{5 \, b c^{8} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{128 \, \sqrt{d^{2}} d^{6}} - \frac{a c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{16 \, \sqrt{d^{2}} d^{4}} + \frac{5 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{6} x}{128 \, d^{6}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a c^{4} x}{16 \, d^{4}} + \frac{5 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{4} x}{64 \, d^{6}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a c^{2} x}{8 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)*x^4,x, algorithm="maxima")

[Out]

1/8*(d^2*x^2 - c^2)^(3/2)*b*x^5/d^2 + 5/48*(d^2*x^2 - c^2)^(3/2)*b*c^2*x^3/d^4 +
1/6*(d^2*x^2 - c^2)^(3/2)*a*x^3/d^2 - 5/128*b*c^8*log(2*d^2*x + 2*sqrt(d^2*x^2
- c^2)*sqrt(d^2))/(sqrt(d^2)*d^6) - 1/16*a*c^6*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^
2)*sqrt(d^2))/(sqrt(d^2)*d^4) + 5/128*sqrt(d^2*x^2 - c^2)*b*c^6*x/d^6 + 1/16*sqr
t(d^2*x^2 - c^2)*a*c^4*x/d^4 + 5/64*(d^2*x^2 - c^2)^(3/2)*b*c^4*x/d^6 + 1/8*(d^2
*x^2 - c^2)^(3/2)*a*c^2*x/d^4

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Fricas [A]  time = 0.651568, size = 976, normalized size = 4.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)*x^4,x, algorithm="fricas")

[Out]

-1/384*(6144*b*d^16*x^16 - 8192*(2*b*c^2*d^14 - a*d^16)*x^14 + 2048*(7*b*c^4*d^1
2 - 11*a*c^2*d^14)*x^12 - 1024*(5*b*c^6*d^10 - 19*a*c^4*d^12)*x^10 + 288*(11*b*c
^8*d^8 - 8*a*c^6*d^10)*x^8 - 192*(17*b*c^10*d^6 + 24*a*c^8*d^8)*x^6 + 248*(5*b*c
^12*d^4 + 8*a*c^10*d^6)*x^4 - 24*(5*b*c^14*d^2 + 8*a*c^12*d^4)*x^2 - (6144*b*d^1
5*x^15 - 1024*(13*b*c^2*d^13 - 8*a*d^15)*x^13 + 768*(11*b*c^4*d^11 - 24*a*c^2*d^
13)*x^11 - 128*(17*b*c^6*d^9 - 88*a*c^4*d^11)*x^9 + 48*(53*b*c^8*d^7 + 32*a*c^6*
d^9)*x^7 - 24*(87*b*c^10*d^5 + 136*a*c^8*d^7)*x^5 + 94*(5*b*c^12*d^3 + 8*a*c^10*
d^5)*x^3 - 3*(5*b*c^14*d + 8*a*c^12*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) - 3*(5*b
*c^16 + 8*a*c^14*d^2 + 128*(5*b*c^8*d^8 + 8*a*c^6*d^10)*x^8 - 256*(5*b*c^10*d^6
+ 8*a*c^8*d^8)*x^6 + 160*(5*b*c^12*d^4 + 8*a*c^10*d^6)*x^4 - 32*(5*b*c^14*d^2 +
8*a*c^12*d^4)*x^2 - 8*(16*(5*b*c^8*d^7 + 8*a*c^6*d^9)*x^7 - 24*(5*b*c^10*d^5 + 8
*a*c^8*d^7)*x^5 + 10*(5*b*c^12*d^3 + 8*a*c^10*d^5)*x^3 - (5*b*c^14*d + 8*a*c^12*
d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))/(1
28*d^15*x^8 - 256*c^2*d^13*x^6 + 160*c^4*d^11*x^4 - 32*c^6*d^9*x^2 + c^8*d^7 - 8
*(16*d^14*x^7 - 24*c^2*d^12*x^5 + 10*c^4*d^10*x^3 - c^6*d^8*x)*sqrt(d*x + c)*sqr
t(d*x - c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.306101, size = 394, normalized size = 1.89 \[ \frac{8 \,{\left (\frac{6 \, c^{6}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{d^{4}} +{\left ({\left (2 \,{\left ({\left (d x + c\right )}{\left (4 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{4}} - \frac{5 \, c}{d^{4}}\right )} + \frac{39 \, c^{2}}{d^{4}}\right )} - \frac{37 \, c^{3}}{d^{4}}\right )}{\left (d x + c\right )} + \frac{31 \, c^{4}}{d^{4}}\right )}{\left (d x + c\right )} - \frac{3 \, c^{5}}{d^{4}}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} a +{\left (\frac{30 \, c^{8}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{d^{6}} +{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (d x + c\right )}{\left (6 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{6}} - \frac{7 \, c}{d^{6}}\right )} + \frac{125 \, c^{2}}{d^{6}}\right )} - \frac{205 \, c^{3}}{d^{6}}\right )}{\left (d x + c\right )} + \frac{795 \, c^{4}}{d^{6}}\right )}{\left (d x + c\right )} - \frac{449 \, c^{5}}{d^{6}}\right )}{\left (d x + c\right )} + \frac{251 \, c^{6}}{d^{6}}\right )}{\left (d x + c\right )} - \frac{15 \, c^{7}}{d^{6}}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} b}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)*x^4,x, algorithm="giac")

[Out]

1/384*(8*(6*c^6*ln(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^4 + ((2*((d*x + c)*(4*
(d*x + c)*((d*x + c)/d^4 - 5*c/d^4) + 39*c^2/d^4) - 37*c^3/d^4)*(d*x + c) + 31*c
^4/d^4)*(d*x + c) - 3*c^5/d^4)*sqrt(d*x + c)*sqrt(d*x - c))*a + (30*c^8*ln(abs(-
sqrt(d*x + c) + sqrt(d*x - c)))/d^6 + ((2*((4*((d*x + c)*(6*(d*x + c)*((d*x + c)
/d^6 - 7*c/d^6) + 125*c^2/d^6) - 205*c^3/d^6)*(d*x + c) + 795*c^4/d^6)*(d*x + c)
- 449*c^5/d^6)*(d*x + c) + 251*c^6/d^6)*(d*x + c) - 15*c^7/d^6)*sqrt(d*x + c)*s
qrt(d*x - c))*b)/d