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

Optimal. Leaf size=164 \[ \frac{c^2 x \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{16 d^6}+\frac{x^3 \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{24 d^4}+\frac{c^4 \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{8 d^7}+\frac{b x^5 \sqrt{d x-c} \sqrt{c+d x}}{6 d^2} \]

[Out]

(c^2*(5*b*c^2 + 6*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(16*d^6) + ((5*b*c^2 +
6*a*d^2)*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x])/(24*d^4) + (b*x^5*Sqrt[-c + d*x]*Sqrt
[c + d*x])/(6*d^2) + (c^4*(5*b*c^2 + 6*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*
x]])/(8*d^7)

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Rubi [A]  time = 0.422232, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{c^2 x \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{16 d^6}+\frac{x^3 \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2+5 b c^2\right )}{24 d^4}+\frac{c^4 \left (6 a d^2+5 b c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{8 d^7}+\frac{b x^5 \sqrt{d x-c} \sqrt{c+d x}}{6 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(5*b*c^2 + 6*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(16*d^6) + ((5*b*c^2 +
6*a*d^2)*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x])/(24*d^4) + (b*x^5*Sqrt[-c + d*x]*Sqrt
[c + d*x])/(6*d^2) + (c^4*(5*b*c^2 + 6*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*
x]])/(8*d^7)

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Rubi in Sympy [A]  time = 26.1904, size = 146, normalized size = 0.89 \[ \frac{b x^{5} \sqrt{- c + d x} \sqrt{c + d x}}{6 d^{2}} + \frac{c^{4} \left (6 a d^{2} + 5 b c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d x}} \right )}}{8 d^{7}} + \frac{c^{2} x \sqrt{- c + d x} \sqrt{c + d x} \left (6 a d^{2} + 5 b c^{2}\right )}{16 d^{6}} + \frac{x^{3} \sqrt{- c + d x} \sqrt{c + d x} \left (6 a d^{2} + 5 b c^{2}\right )}{24 d^{4}} \]

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*sqrt(-c + d*x)*sqrt(c + d*x)/(6*d**2) + c**4*(6*a*d**2 + 5*b*c**2)*atanh(
sqrt(c + d*x)/sqrt(-c + d*x))/(8*d**7) + c**2*x*sqrt(-c + d*x)*sqrt(c + d*x)*(6*
a*d**2 + 5*b*c**2)/(16*d**6) + x**3*sqrt(-c + d*x)*sqrt(c + d*x)*(6*a*d**2 + 5*b
*c**2)/(24*d**4)

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Mathematica [A]  time = 0.170729, size = 123, normalized size = 0.75 \[ \frac{3 \left (6 a c^4 d^2+5 b c^6\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )+d x \sqrt{d x-c} \sqrt{c+d x} \left (6 a d^2 \left (3 c^2+2 d^2 x^2\right )+b \left (15 c^4+10 c^2 d^2 x^2+8 d^4 x^4\right )\right )}{48 d^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.04, size = 240, normalized size = 1.5 \[{\frac{{\it csgn} \left ( d \right ) }{48\,{d}^{7}}\sqrt{dx-c}\sqrt{dx+c} \left ( 8\,{\it csgn} \left ( d \right ){x}^{5}b{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+12\,{\it csgn} \left ( d \right ){x}^{3}a{d}^{5}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+10\,{\it csgn} \left ( d \right ){x}^{3}b{c}^{2}{d}^{3}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}+18\,a{c}^{2}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) +15\,b{c}^{4}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d+18\,a{c}^{4}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){d}^{2}+15\,b{c}^{6}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

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]

1/48*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(8*csgn(d)*x^5*b*d^5*(d^2*x^2-c^2)^(1/2)+12*csg
n(d)*x^3*a*d^5*(d^2*x^2-c^2)^(1/2)+10*csgn(d)*x^3*b*c^2*d^3*(d^2*x^2-c^2)^(1/2)+
18*a*c^2*x*(d^2*x^2-c^2)^(1/2)*d^3*csgn(d)+15*b*c^4*x*(d^2*x^2-c^2)^(1/2)*csgn(d
)*d+18*a*c^4*ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))*d^2+15*b*c^6*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 = 1.39756, size = 289, normalized size = 1.76 \[ \frac{\sqrt{d^{2} x^{2} - c^{2}} b x^{5}}{6 \, d^{2}} + \frac{5 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{2} x^{3}}{24 \, d^{4}} + \frac{\sqrt{d^{2} x^{2} - c^{2}} a x^{3}}{4 \, d^{2}} + \frac{5 \, b c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{16 \, \sqrt{d^{2}} d^{6}} + \frac{3 \, a c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{8 \, \sqrt{d^{2}} d^{4}} + \frac{5 \, \sqrt{d^{2} x^{2} - c^{2}} b c^{4} x}{16 \, d^{6}} + \frac{3 \, \sqrt{d^{2} x^{2} - c^{2}} a c^{2} x}{8 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(d^2*x^2 - c^2)*b*x^5/d^2 + 5/24*sqrt(d^2*x^2 - c^2)*b*c^2*x^3/d^4 + 1/4
*sqrt(d^2*x^2 - c^2)*a*x^3/d^2 + 5/16*b*c^6*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*
sqrt(d^2))/(sqrt(d^2)*d^6) + 3/8*a*c^4*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*sqrt(
d^2))/(sqrt(d^2)*d^4) + 5/16*sqrt(d^2*x^2 - c^2)*b*c^4*x/d^6 + 3/8*sqrt(d^2*x^2
- c^2)*a*c^2*x/d^4

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Fricas [A]  time = 0.361266, size = 734, normalized size = 4.48 \[ -\frac{256 \, b d^{12} x^{12} - 192 \,{\left (b c^{2} d^{10} - 2 \, a d^{12}\right )} x^{10} + 48 \,{\left (3 \, b c^{4} d^{8} - 4 \, a c^{2} d^{10}\right )} x^{8} - 4 \,{\left (157 \, b c^{6} d^{6} + 174 \, a c^{4} d^{8}\right )} x^{6} + 102 \,{\left (5 \, b c^{8} d^{4} + 6 \, a c^{6} d^{6}\right )} x^{4} - 18 \,{\left (5 \, b c^{10} d^{2} + 6 \, a c^{8} d^{4}\right )} x^{2} -{\left (256 \, b d^{11} x^{11} + 144 \, b c^{4} d^{7} x^{7} - 64 \,{\left (b c^{2} d^{9} - 6 \, a d^{11}\right )} x^{9} - 4 \,{\left (137 \, b c^{6} d^{5} + 162 \, a c^{4} d^{7}\right )} x^{5} + 52 \,{\left (5 \, b c^{8} d^{3} + 6 \, a c^{6} d^{5}\right )} x^{3} - 3 \,{\left (5 \, b c^{10} d + 6 \, a c^{8} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c} - 3 \,{\left (5 \, b c^{12} + 6 \, a c^{10} d^{2} - 32 \,{\left (5 \, b c^{6} d^{6} + 6 \, a c^{4} d^{8}\right )} x^{6} + 48 \,{\left (5 \, b c^{8} d^{4} + 6 \, a c^{6} d^{6}\right )} x^{4} - 18 \,{\left (5 \, b c^{10} d^{2} + 6 \, a c^{8} d^{4}\right )} x^{2} + 2 \,{\left (16 \,{\left (5 \, b c^{6} d^{5} + 6 \, a c^{4} d^{7}\right )} x^{5} - 16 \,{\left (5 \, b c^{8} d^{3} + 6 \, a c^{6} d^{5}\right )} x^{3} + 3 \,{\left (5 \, b c^{10} d + 6 \, a c^{8} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{48 \,{\left (32 \, d^{13} x^{6} - 48 \, c^{2} d^{11} x^{4} + 18 \, c^{4} d^{9} x^{2} - c^{6} d^{7} - 2 \,{\left (16 \, d^{12} x^{5} - 16 \, c^{2} d^{10} x^{3} + 3 \, c^{4} d^{8} x\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(256*b*d^12*x^12 - 192*(b*c^2*d^10 - 2*a*d^12)*x^10 + 48*(3*b*c^4*d^8 - 4*
a*c^2*d^10)*x^8 - 4*(157*b*c^6*d^6 + 174*a*c^4*d^8)*x^6 + 102*(5*b*c^8*d^4 + 6*a
*c^6*d^6)*x^4 - 18*(5*b*c^10*d^2 + 6*a*c^8*d^4)*x^2 - (256*b*d^11*x^11 + 144*b*c
^4*d^7*x^7 - 64*(b*c^2*d^9 - 6*a*d^11)*x^9 - 4*(137*b*c^6*d^5 + 162*a*c^4*d^7)*x
^5 + 52*(5*b*c^8*d^3 + 6*a*c^6*d^5)*x^3 - 3*(5*b*c^10*d + 6*a*c^8*d^3)*x)*sqrt(d
*x + c)*sqrt(d*x - c) - 3*(5*b*c^12 + 6*a*c^10*d^2 - 32*(5*b*c^6*d^6 + 6*a*c^4*d
^8)*x^6 + 48*(5*b*c^8*d^4 + 6*a*c^6*d^6)*x^4 - 18*(5*b*c^10*d^2 + 6*a*c^8*d^4)*x
^2 + 2*(16*(5*b*c^6*d^5 + 6*a*c^4*d^7)*x^5 - 16*(5*b*c^8*d^3 + 6*a*c^6*d^5)*x^3
+ 3*(5*b*c^10*d + 6*a*c^8*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))*log(-d*x + sqrt(d
*x + c)*sqrt(d*x - c)))/(32*d^13*x^6 - 48*c^2*d^11*x^4 + 18*c^4*d^9*x^2 - c^6*d^
7 - 2*(16*d^12*x^5 - 16*c^2*d^10*x^3 + 3*c^4*d^8*x)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.258263, size = 246, normalized size = 1.5 \[ -\frac{{\left (33 \, b c^{5} d^{36} + 30 \, a c^{3} d^{38} -{\left (85 \, b c^{4} d^{36} + 54 \, a c^{2} d^{38} - 2 \,{\left (55 \, b c^{3} d^{36} + 18 \, a c d^{38} -{\left (45 \, b c^{2} d^{36} + 6 \, a d^{38} + 4 \,{\left ({\left (d x + c\right )} b d^{36} - 5 \, b c d^{36}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )}{\left (d x + c\right )}\right )} \sqrt{d x + c} \sqrt{d x - c} + 6 \,{\left (5 \, b c^{6} d^{36} + 6 \, a c^{4} d^{38}\right )}{\rm ln}\left ({\left | -\sqrt{d x + c} + \sqrt{d x - c} \right |}\right )}{34603008 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/34603008*((33*b*c^5*d^36 + 30*a*c^3*d^38 - (85*b*c^4*d^36 + 54*a*c^2*d^38 - 2
*(55*b*c^3*d^36 + 18*a*c*d^38 - (45*b*c^2*d^36 + 6*a*d^38 + 4*((d*x + c)*b*d^36
- 5*b*c*d^36)*(d*x + c))*(d*x + c))*(d*x + c))*(d*x + c))*sqrt(d*x + c)*sqrt(d*x
- c) + 6*(5*b*c^6*d^36 + 6*a*c^4*d^38)*ln(abs(-sqrt(d*x + c) + sqrt(d*x - c))))
/d