3.522 \(\int x^4 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=188 \[ \frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}-\frac{3 a^3 x \sqrt{a+b x^2} (2 A b-a B)}{256 b^3}+\frac{a^2 x^3 \sqrt{a+b x^2} (2 A b-a B)}{128 b^2}+\frac{x^5 \left (a+b x^2\right )^{3/2} (2 A b-a B)}{16 b}+\frac{a x^5 \sqrt{a+b x^2} (2 A b-a B)}{32 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b} \]

[Out]

(-3*a^3*(2*A*b - a*B)*x*Sqrt[a + b*x^2])/(256*b^3) + (a^2*(2*A*b - a*B)*x^3*Sqrt
[a + b*x^2])/(128*b^2) + (a*(2*A*b - a*B)*x^5*Sqrt[a + b*x^2])/(32*b) + ((2*A*b
- a*B)*x^5*(a + b*x^2)^(3/2))/(16*b) + (B*x^5*(a + b*x^2)^(5/2))/(10*b) + (3*a^4
*(2*A*b - a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(7/2))

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Rubi [A]  time = 0.28575, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}-\frac{3 a^3 x \sqrt{a+b x^2} (2 A b-a B)}{256 b^3}+\frac{a^2 x^3 \sqrt{a+b x^2} (2 A b-a B)}{128 b^2}+\frac{x^5 \left (a+b x^2\right )^{3/2} (2 A b-a B)}{16 b}+\frac{a x^5 \sqrt{a+b x^2} (2 A b-a B)}{32 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^2)^(3/2)*(A + B*x^2),x]

[Out]

(-3*a^3*(2*A*b - a*B)*x*Sqrt[a + b*x^2])/(256*b^3) + (a^2*(2*A*b - a*B)*x^3*Sqrt
[a + b*x^2])/(128*b^2) + (a*(2*A*b - a*B)*x^5*Sqrt[a + b*x^2])/(32*b) + ((2*A*b
- a*B)*x^5*(a + b*x^2)^(3/2))/(16*b) + (B*x^5*(a + b*x^2)^(5/2))/(10*b) + (3*a^4
*(2*A*b - a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(7/2))

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Rubi in Sympy [A]  time = 27.8905, size = 170, normalized size = 0.9 \[ \frac{B x^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}}{10 b} + \frac{3 a^{4} \left (2 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{7}{2}}} - \frac{3 a^{3} x \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{256 b^{3}} + \frac{a^{2} x^{3} \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{128 b^{2}} + \frac{a x^{5} \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{32 b} + \frac{x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - B a\right )}{16 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**(3/2)*(B*x**2+A),x)

[Out]

B*x**5*(a + b*x**2)**(5/2)/(10*b) + 3*a**4*(2*A*b - B*a)*atanh(sqrt(b)*x/sqrt(a
+ b*x**2))/(256*b**(7/2)) - 3*a**3*x*sqrt(a + b*x**2)*(2*A*b - B*a)/(256*b**3) +
a**2*x**3*sqrt(a + b*x**2)*(2*A*b - B*a)/(128*b**2) + a*x**5*sqrt(a + b*x**2)*(
2*A*b - B*a)/(32*b) + x**5*(a + b*x**2)**(3/2)*(2*A*b - B*a)/(16*b)

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Mathematica [A]  time = 0.157687, size = 140, normalized size = 0.74 \[ \sqrt{a+b x^2} \left (\frac{3 a^3 x (a B-2 A b)}{256 b^3}-\frac{a^2 x^3 (a B-2 A b)}{128 b^2}+\frac{1}{80} x^7 (11 a B+10 A b)+\frac{a x^5 (a B+30 A b)}{160 b}+\frac{1}{10} b B x^9\right )-\frac{3 a^4 (a B-2 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{256 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^2)^(3/2)*(A + B*x^2),x]

[Out]

Sqrt[a + b*x^2]*((3*a^3*(-2*A*b + a*B)*x)/(256*b^3) - (a^2*(-2*A*b + a*B)*x^3)/(
128*b^2) + (a*(30*A*b + a*B)*x^5)/(160*b) + ((10*A*b + 11*a*B)*x^7)/80 + (b*B*x^
9)/10) - (3*a^4*(-2*A*b + a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(256*b^(7/2))

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Maple [A]  time = 0.012, size = 219, normalized size = 1.2 \[{\frac{A{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aAx}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Ax}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,A{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{B{x}^{5}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ba{x}^{3}}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bx{a}^{2}}{32\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{a}^{3}x}{128\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{a}^{4}x}{256\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{3\,B{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^(3/2)*(B*x^2+A),x)

[Out]

1/8*A*x^3*(b*x^2+a)^(5/2)/b-1/16*A*a/b^2*x*(b*x^2+a)^(5/2)+1/64*A*a^2/b^2*x*(b*x
^2+a)^(3/2)+3/128*A*a^3/b^2*x*(b*x^2+a)^(1/2)+3/128*A*a^4/b^(5/2)*ln(x*b^(1/2)+(
b*x^2+a)^(1/2))+1/10*B*x^5*(b*x^2+a)^(5/2)/b-1/16*B*a/b^2*x^3*(b*x^2+a)^(5/2)+1/
32*B*a^2/b^3*x*(b*x^2+a)^(5/2)-1/128*B*a^3/b^3*x*(b*x^2+a)^(3/2)-3/256*B*a^4/b^3
*x*(b*x^2+a)^(1/2)-3/256*B*a^5/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.367371, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (128 \, B b^{4} x^{9} + 16 \,{\left (11 \, B a b^{3} + 10 \, A b^{4}\right )} x^{7} + 8 \,{\left (B a^{2} b^{2} + 30 \, A a b^{3}\right )} x^{5} - 10 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{3} + 15 \,{\left (B a^{4} - 2 \, A a^{3} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2560 \, b^{\frac{7}{2}}}, \frac{{\left (128 \, B b^{4} x^{9} + 16 \,{\left (11 \, B a b^{3} + 10 \, A b^{4}\right )} x^{7} + 8 \,{\left (B a^{2} b^{2} + 30 \, A a b^{3}\right )} x^{5} - 10 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{3} + 15 \,{\left (B a^{4} - 2 \, A a^{3} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 15 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{1280 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*x^4,x, algorithm="fricas")

[Out]

[1/2560*(2*(128*B*b^4*x^9 + 16*(11*B*a*b^3 + 10*A*b^4)*x^7 + 8*(B*a^2*b^2 + 30*A
*a*b^3)*x^5 - 10*(B*a^3*b - 2*A*a^2*b^2)*x^3 + 15*(B*a^4 - 2*A*a^3*b)*x)*sqrt(b*
x^2 + a)*sqrt(b) - 15*(B*a^5 - 2*A*a^4*b)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2
+ a)*sqrt(b)))/b^(7/2), 1/1280*((128*B*b^4*x^9 + 16*(11*B*a*b^3 + 10*A*b^4)*x^7
+ 8*(B*a^2*b^2 + 30*A*a*b^3)*x^5 - 10*(B*a^3*b - 2*A*a^2*b^2)*x^3 + 15*(B*a^4 -
2*A*a^3*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 15*(B*a^5 - 2*A*a^4*b)*arctan(sqrt(-b)*
x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 112.338, size = 345, normalized size = 1.84 \[ - \frac{3 A a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 A a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 A \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{A b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{\frac{9}{2}} x}{256 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{\frac{7}{2}} x^{3}}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{5}}{640 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 B a^{\frac{3}{2}} x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{19 B \sqrt{a} b x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 B a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{7}{2}}} + \frac{B b^{2} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)**(3/2)*(B*x**2+A),x)

[Out]

-3*A*a**(7/2)*x/(128*b**2*sqrt(1 + b*x**2/a)) - A*a**(5/2)*x**3/(128*b*sqrt(1 +
b*x**2/a)) + 13*A*a**(3/2)*x**5/(64*sqrt(1 + b*x**2/a)) + 5*A*sqrt(a)*b*x**7/(16
*sqrt(1 + b*x**2/a)) + 3*A*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(5/2)) + A*b**2
*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a)) + 3*B*a**(9/2)*x/(256*b**3*sqrt(1 + b*x**2/
a)) + B*a**(7/2)*x**3/(256*b**2*sqrt(1 + b*x**2/a)) - B*a**(5/2)*x**5/(640*b*sqr
t(1 + b*x**2/a)) + 23*B*a**(3/2)*x**7/(160*sqrt(1 + b*x**2/a)) + 19*B*sqrt(a)*b*
x**9/(80*sqrt(1 + b*x**2/a)) - 3*B*a**5*asinh(sqrt(b)*x/sqrt(a))/(256*b**(7/2))
+ B*b**2*x**11/(10*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.233907, size = 215, normalized size = 1.14 \[ \frac{1}{1280} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, B b x^{2} + \frac{11 \, B a b^{8} + 10 \, A b^{9}}{b^{8}}\right )} x^{2} + \frac{B a^{2} b^{7} + 30 \, A a b^{8}}{b^{8}}\right )} x^{2} - \frac{5 \,{\left (B a^{3} b^{6} - 2 \, A a^{2} b^{7}\right )}}{b^{8}}\right )} x^{2} + \frac{15 \,{\left (B a^{4} b^{5} - 2 \, A a^{3} b^{6}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x + \frac{3 \,{\left (B a^{5} - 2 \, A a^{4} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*x^4,x, algorithm="giac")

[Out]

1/1280*(2*(4*(2*(8*B*b*x^2 + (11*B*a*b^8 + 10*A*b^9)/b^8)*x^2 + (B*a^2*b^7 + 30*
A*a*b^8)/b^8)*x^2 - 5*(B*a^3*b^6 - 2*A*a^2*b^7)/b^8)*x^2 + 15*(B*a^4*b^5 - 2*A*a
^3*b^6)/b^8)*sqrt(b*x^2 + a)*x + 3/256*(B*a^5 - 2*A*a^4*b)*ln(abs(-sqrt(b)*x + s
qrt(b*x^2 + a)))/b^(7/2)