3.541 \(\int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=188 \[ -\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{a^3 x \sqrt{a+b x^2} (10 A b-3 a B)}{256 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2} (10 A b-3 a B)}{128 b}+\frac{a x^3 \left (a+b x^2\right )^{3/2} (10 A b-3 a B)}{96 b}+\frac{x^3 \left (a+b x^2\right )^{5/2} (10 A b-3 a B)}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]

[Out]

(a^3*(10*A*b - 3*a*B)*x*Sqrt[a + b*x^2])/(256*b^2) + (a^2*(10*A*b - 3*a*B)*x^3*S
qrt[a + b*x^2])/(128*b) + (a*(10*A*b - 3*a*B)*x^3*(a + b*x^2)^(3/2))/(96*b) + ((
10*A*b - 3*a*B)*x^3*(a + b*x^2)^(5/2))/(80*b) + (B*x^3*(a + b*x^2)^(7/2))/(10*b)
- (a^4*(10*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))

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Rubi [A]  time = 0.251955, 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 \[ -\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{a^3 x \sqrt{a+b x^2} (10 A b-3 a B)}{256 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2} (10 A b-3 a B)}{128 b}+\frac{a x^3 \left (a+b x^2\right )^{3/2} (10 A b-3 a B)}{96 b}+\frac{x^3 \left (a+b x^2\right )^{5/2} (10 A b-3 a B)}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

(a^3*(10*A*b - 3*a*B)*x*Sqrt[a + b*x^2])/(256*b^2) + (a^2*(10*A*b - 3*a*B)*x^3*S
qrt[a + b*x^2])/(128*b) + (a*(10*A*b - 3*a*B)*x^3*(a + b*x^2)^(3/2))/(96*b) + ((
10*A*b - 3*a*B)*x^3*(a + b*x^2)^(5/2))/(80*b) + (B*x^3*(a + b*x^2)^(7/2))/(10*b)
- (a^4*(10*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))

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Rubi in Sympy [A]  time = 29.735, size = 173, normalized size = 0.92 \[ \frac{B x^{3} \left (a + b x^{2}\right )^{\frac{7}{2}}}{10 b} - \frac{a^{4} \left (10 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{5}{2}}} + \frac{a^{3} x \sqrt{a + b x^{2}} \left (10 A b - 3 B a\right )}{256 b^{2}} + \frac{a^{2} x^{3} \sqrt{a + b x^{2}} \left (10 A b - 3 B a\right )}{128 b} + \frac{a x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (10 A b - 3 B a\right )}{96 b} + \frac{x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}} \left (10 A b - 3 B a\right )}{80 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

B*x**3*(a + b*x**2)**(7/2)/(10*b) - a**4*(10*A*b - 3*B*a)*atanh(sqrt(b)*x/sqrt(a
+ b*x**2))/(256*b**(5/2)) + a**3*x*sqrt(a + b*x**2)*(10*A*b - 3*B*a)/(256*b**2)
+ a**2*x**3*sqrt(a + b*x**2)*(10*A*b - 3*B*a)/(128*b) + a*x**3*(a + b*x**2)**(3
/2)*(10*A*b - 3*B*a)/(96*b) + x**3*(a + b*x**2)**(5/2)*(10*A*b - 3*B*a)/(80*b)

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Mathematica [A]  time = 0.167115, size = 144, normalized size = 0.77 \[ \frac{a^4 (3 a B-10 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{256 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{a^3 x (3 a B-10 A b)}{256 b^2}+\frac{a^2 x^3 (3 a B+118 A b)}{384 b}+\frac{1}{80} b x^7 (21 a B+10 A b)+\frac{1}{480} a x^5 (93 a B+170 A b)+\frac{1}{10} b^2 B x^9\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

Sqrt[a + b*x^2]*(-(a^3*(-10*A*b + 3*a*B)*x)/(256*b^2) + (a^2*(118*A*b + 3*a*B)*x
^3)/(384*b) + (a*(170*A*b + 93*a*B)*x^5)/480 + (b*(10*A*b + 21*a*B)*x^7)/80 + (b
^2*B*x^9)/10) + (a^4*(-10*A*b + 3*a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(256*
b^(5/2))

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Maple [A]  time = 0.01, size = 215, normalized size = 1.1 \[{\frac{Ax}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{aAx}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}Ax}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,A{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Bxa}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bx{a}^{2}}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,B{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,B{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(5/2)*(B*x^2+A),x)

[Out]

1/8*A*x*(b*x^2+a)^(7/2)/b-1/48*A*a/b*x*(b*x^2+a)^(5/2)-5/192*A*a^2/b*x*(b*x^2+a)
^(3/2)-5/128*A*a^3/b*x*(b*x^2+a)^(1/2)-5/128*A*a^4/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a
)^(1/2))+1/10*B*x^3*(b*x^2+a)^(7/2)/b-3/80*B*a/b^2*x*(b*x^2+a)^(7/2)+1/160*B*a^2
/b^2*x*(b*x^2+a)^(5/2)+1/128*B*a^3/b^2*x*(b*x^2+a)^(3/2)+3/256*B*a^4/b^2*x*(b*x^
2+a)^(1/2)+3/256*B*a^5/b^(5/2)*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)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.365279, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (384 \, B b^{4} x^{9} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} x^{7} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} x^{5} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} x^{3} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{7680 \, b^{\frac{5}{2}}}, \frac{{\left (384 \, B b^{4} x^{9} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} x^{7} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} x^{5} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} x^{3} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3840 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x^2,x, algorithm="fricas")

[Out]

[1/7680*(2*(384*B*b^4*x^9 + 48*(21*B*a*b^3 + 10*A*b^4)*x^7 + 8*(93*B*a^2*b^2 + 1
70*A*a*b^3)*x^5 + 10*(3*B*a^3*b + 118*A*a^2*b^2)*x^3 - 15*(3*B*a^4 - 10*A*a^3*b)
*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(3*B*a^5 - 10*A*a^4*b)*log(2*sqrt(b*x^2 + a)*b*
x - (2*b*x^2 + a)*sqrt(b)))/b^(5/2), 1/3840*((384*B*b^4*x^9 + 48*(21*B*a*b^3 + 1
0*A*b^4)*x^7 + 8*(93*B*a^2*b^2 + 170*A*a*b^3)*x^5 + 10*(3*B*a^3*b + 118*A*a^2*b^
2)*x^3 - 15*(3*B*a^4 - 10*A*a^3*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 15*(3*B*a^5 - 1
0*A*a^4*b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 138.628, size = 348, normalized size = 1.85 \[ \frac{5 A a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 A a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 A a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 A \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 A a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{A b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 B a^{\frac{9}{2}} x}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{7}{2}} x^{3}}{256 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{129 B a^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{73 B a^{\frac{3}{2}} b x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 B \sqrt{a} b^{2} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{5}{2}}} + \frac{B b^{3} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

5*A*a**(7/2)*x/(128*b*sqrt(1 + b*x**2/a)) + 133*A*a**(5/2)*x**3/(384*sqrt(1 + b*
x**2/a)) + 127*A*a**(3/2)*b*x**5/(192*sqrt(1 + b*x**2/a)) + 23*A*sqrt(a)*b**2*x*
*7/(48*sqrt(1 + b*x**2/a)) - 5*A*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(3/2)) +
A*b**3*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a)) - 3*B*a**(9/2)*x/(256*b**2*sqrt(1 + b
*x**2/a)) - B*a**(7/2)*x**3/(256*b*sqrt(1 + b*x**2/a)) + 129*B*a**(5/2)*x**5/(64
0*sqrt(1 + b*x**2/a)) + 73*B*a**(3/2)*b*x**7/(160*sqrt(1 + b*x**2/a)) + 29*B*sqr
t(a)*b**2*x**9/(80*sqrt(1 + b*x**2/a)) + 3*B*a**5*asinh(sqrt(b)*x/sqrt(a))/(256*
b**(5/2)) + B*b**3*x**11/(10*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.250301, size = 223, normalized size = 1.19 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, B b^{2} x^{2} + \frac{21 \, B a b^{9} + 10 \, A b^{10}}{b^{8}}\right )} x^{2} + \frac{93 \, B a^{2} b^{8} + 170 \, A a b^{9}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (3 \, B a^{3} b^{7} + 118 \, A a^{2} b^{8}\right )}}{b^{8}}\right )} x^{2} - \frac{15 \,{\left (3 \, B a^{4} b^{6} - 10 \, A a^{3} b^{7}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x^2,x, algorithm="giac")

[Out]

1/3840*(2*(4*(6*(8*B*b^2*x^2 + (21*B*a*b^9 + 10*A*b^10)/b^8)*x^2 + (93*B*a^2*b^8
+ 170*A*a*b^9)/b^8)*x^2 + 5*(3*B*a^3*b^7 + 118*A*a^2*b^8)/b^8)*x^2 - 15*(3*B*a^
4*b^6 - 10*A*a^3*b^7)/b^8)*sqrt(b*x^2 + a)*x - 1/256*(3*B*a^5 - 10*A*a^4*b)*ln(a
bs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)