3.539 \(\int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=221 \[ \frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{7/2}}-\frac{a^4 x \sqrt{a+b x^2} (12 A b-5 a B)}{1024 b^3}+\frac{a^3 x^3 \sqrt{a+b x^2} (12 A b-5 a B)}{1536 b^2}+\frac{a^2 x^5 \sqrt{a+b x^2} (12 A b-5 a B)}{384 b}+\frac{a x^5 \left (a+b x^2\right )^{3/2} (12 A b-5 a B)}{192 b}+\frac{x^5 \left (a+b x^2\right )^{5/2} (12 A b-5 a B)}{120 b}+\frac{B x^5 \left (a+b x^2\right )^{7/2}}{12 b} \]

[Out]

-(a^4*(12*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(1024*b^3) + (a^3*(12*A*b - 5*a*B)*x^3
*Sqrt[a + b*x^2])/(1536*b^2) + (a^2*(12*A*b - 5*a*B)*x^5*Sqrt[a + b*x^2])/(384*b
) + (a*(12*A*b - 5*a*B)*x^5*(a + b*x^2)^(3/2))/(192*b) + ((12*A*b - 5*a*B)*x^5*(
a + b*x^2)^(5/2))/(120*b) + (B*x^5*(a + b*x^2)^(7/2))/(12*b) + (a^5*(12*A*b - 5*
a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))

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Rubi [A]  time = 0.314075, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{7/2}}-\frac{a^4 x \sqrt{a+b x^2} (12 A b-5 a B)}{1024 b^3}+\frac{a^3 x^3 \sqrt{a+b x^2} (12 A b-5 a B)}{1536 b^2}+\frac{a^2 x^5 \sqrt{a+b x^2} (12 A b-5 a B)}{384 b}+\frac{a x^5 \left (a+b x^2\right )^{3/2} (12 A b-5 a B)}{192 b}+\frac{x^5 \left (a+b x^2\right )^{5/2} (12 A b-5 a B)}{120 b}+\frac{B x^5 \left (a+b x^2\right )^{7/2}}{12 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

-(a^4*(12*A*b - 5*a*B)*x*Sqrt[a + b*x^2])/(1024*b^3) + (a^3*(12*A*b - 5*a*B)*x^3
*Sqrt[a + b*x^2])/(1536*b^2) + (a^2*(12*A*b - 5*a*B)*x^5*Sqrt[a + b*x^2])/(384*b
) + (a*(12*A*b - 5*a*B)*x^5*(a + b*x^2)^(3/2))/(192*b) + ((12*A*b - 5*a*B)*x^5*(
a + b*x^2)^(5/2))/(120*b) + (B*x^5*(a + b*x^2)^(7/2))/(12*b) + (a^5*(12*A*b - 5*
a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))

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Rubi in Sympy [A]  time = 33.5531, size = 206, normalized size = 0.93 \[ \frac{B x^{5} \left (a + b x^{2}\right )^{\frac{7}{2}}}{12 b} + \frac{a^{5} \left (12 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{1024 b^{\frac{7}{2}}} - \frac{a^{4} x \sqrt{a + b x^{2}} \left (12 A b - 5 B a\right )}{1024 b^{3}} + \frac{a^{3} x^{3} \sqrt{a + b x^{2}} \left (12 A b - 5 B a\right )}{1536 b^{2}} + \frac{a^{2} x^{5} \sqrt{a + b x^{2}} \left (12 A b - 5 B a\right )}{384 b} + \frac{a x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (12 A b - 5 B a\right )}{192 b} + \frac{x^{5} \left (a + b x^{2}\right )^{\frac{5}{2}} \left (12 A b - 5 B a\right )}{120 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

B*x**5*(a + b*x**2)**(7/2)/(12*b) + a**5*(12*A*b - 5*B*a)*atanh(sqrt(b)*x/sqrt(a
+ b*x**2))/(1024*b**(7/2)) - a**4*x*sqrt(a + b*x**2)*(12*A*b - 5*B*a)/(1024*b**
3) + a**3*x**3*sqrt(a + b*x**2)*(12*A*b - 5*B*a)/(1536*b**2) + a**2*x**5*sqrt(a
+ b*x**2)*(12*A*b - 5*B*a)/(384*b) + a*x**5*(a + b*x**2)**(3/2)*(12*A*b - 5*B*a)
/(192*b) + x**5*(a + b*x**2)**(5/2)*(12*A*b - 5*B*a)/(120*b)

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Mathematica [A]  time = 0.187, size = 166, normalized size = 0.75 \[ \sqrt{a+b x^2} \left (\frac{a^4 x (5 a B-12 A b)}{1024 b^3}-\frac{a^3 x^3 (5 a B-12 A b)}{1536 b^2}+\frac{a^2 x^5 (5 a B+372 A b)}{1920 b}+\frac{1}{120} b x^9 (25 a B+12 A b)+\frac{3}{320} a x^7 (15 a B+28 A b)+\frac{1}{12} b^2 B x^{11}\right )-\frac{a^5 (5 a B-12 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{1024 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

Sqrt[a + b*x^2]*((a^4*(-12*A*b + 5*a*B)*x)/(1024*b^3) - (a^3*(-12*A*b + 5*a*B)*x
^3)/(1536*b^2) + (a^2*(372*A*b + 5*a*B)*x^5)/(1920*b) + (3*a*(28*A*b + 15*a*B)*x
^7)/320 + (b*(12*A*b + 25*a*B)*x^9)/120 + (b^2*B*x^11)/12) - (a^5*(-12*A*b + 5*a
*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(1024*b^(7/2))

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Maple [A]  time = 0.012, size = 257, normalized size = 1.2 \[{\frac{A{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,aAx}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}Ax}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,A{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{B{x}^{5}}{12\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Ba{x}^{3}}{24\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bx{a}^{2}}{64\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{a}^{3}x}{384\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,B{a}^{4}x}{1536\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bx{a}^{5}}{1024\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{6}}{1024}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^(5/2)*(B*x^2+A),x)

[Out]

1/10*A*x^3*(b*x^2+a)^(7/2)/b-3/80*A*a/b^2*x*(b*x^2+a)^(7/2)+1/160*A*a^2/b^2*x*(b
*x^2+a)^(5/2)+1/128*A*a^3/b^2*x*(b*x^2+a)^(3/2)+3/256*A*a^4/b^2*x*(b*x^2+a)^(1/2
)+3/256*A*a^5/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/12*B*x^5*(b*x^2+a)^(7/2)/b
-1/24*B*a/b^2*x^3*(b*x^2+a)^(7/2)+1/64*B*a^2/b^3*x*(b*x^2+a)^(7/2)-1/384*B*a^3/b
^3*x*(b*x^2+a)^(5/2)-5/1536*B*a^4/b^3*x*(b*x^2+a)^(3/2)-5/1024*B*a^5/b^3*x*(b*x^
2+a)^(1/2)-5/1024*B*a^6/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)^(5/2)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.552754, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, B b^{5} x^{11} + 128 \,{\left (25 \, B a b^{4} + 12 \, A b^{5}\right )} x^{9} + 144 \,{\left (15 \, B a^{2} b^{3} + 28 \, A a b^{4}\right )} x^{7} + 8 \,{\left (5 \, B a^{3} b^{2} + 372 \, A a^{2} b^{3}\right )} x^{5} - 10 \,{\left (5 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x^{3} + 15 \,{\left (5 \, B a^{5} - 12 \, A a^{4} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{30720 \, b^{\frac{7}{2}}}, \frac{{\left (1280 \, B b^{5} x^{11} + 128 \,{\left (25 \, B a b^{4} + 12 \, A b^{5}\right )} x^{9} + 144 \,{\left (15 \, B a^{2} b^{3} + 28 \, A a b^{4}\right )} x^{7} + 8 \,{\left (5 \, B a^{3} b^{2} + 372 \, A a^{2} b^{3}\right )} x^{5} - 10 \,{\left (5 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x^{3} + 15 \,{\left (5 \, B a^{5} - 12 \, A a^{4} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 15 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{15360 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x^4,x, algorithm="fricas")

[Out]

[1/30720*(2*(1280*B*b^5*x^11 + 128*(25*B*a*b^4 + 12*A*b^5)*x^9 + 144*(15*B*a^2*b
^3 + 28*A*a*b^4)*x^7 + 8*(5*B*a^3*b^2 + 372*A*a^2*b^3)*x^5 - 10*(5*B*a^4*b - 12*
A*a^3*b^2)*x^3 + 15*(5*B*a^5 - 12*A*a^4*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(5*B*
a^6 - 12*A*a^5*b)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(7/2),
1/15360*((1280*B*b^5*x^11 + 128*(25*B*a*b^4 + 12*A*b^5)*x^9 + 144*(15*B*a^2*b^3
+ 28*A*a*b^4)*x^7 + 8*(5*B*a^3*b^2 + 372*A*a^2*b^3)*x^5 - 10*(5*B*a^4*b - 12*A*a
^3*b^2)*x^3 + 15*(5*B*a^5 - 12*A*a^4*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 15*(5*B*a^
6 - 12*A*a^5*b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^3)]

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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)**(5/2)*(B*x**2+A),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247971, size = 263, normalized size = 1.19 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B b^{2} x^{2} + \frac{25 \, B a b^{11} + 12 \, A b^{12}}{b^{10}}\right )} x^{2} + \frac{9 \,{\left (15 \, B a^{2} b^{10} + 28 \, A a b^{11}\right )}}{b^{10}}\right )} x^{2} + \frac{5 \, B a^{3} b^{9} + 372 \, A a^{2} b^{10}}{b^{10}}\right )} x^{2} - \frac{5 \,{\left (5 \, B a^{4} b^{8} - 12 \, A a^{3} b^{9}\right )}}{b^{10}}\right )} x^{2} + \frac{15 \,{\left (5 \, B a^{5} b^{7} - 12 \, A a^{4} b^{8}\right )}}{b^{10}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{1024 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x^4,x, algorithm="giac")

[Out]

1/15360*(2*(4*(2*(8*(10*B*b^2*x^2 + (25*B*a*b^11 + 12*A*b^12)/b^10)*x^2 + 9*(15*
B*a^2*b^10 + 28*A*a*b^11)/b^10)*x^2 + (5*B*a^3*b^9 + 372*A*a^2*b^10)/b^10)*x^2 -
5*(5*B*a^4*b^8 - 12*A*a^3*b^9)/b^10)*x^2 + 15*(5*B*a^5*b^7 - 12*A*a^4*b^8)/b^10
)*sqrt(b*x^2 + a)*x + 1/1024*(5*B*a^6 - 12*A*a^5*b)*ln(abs(-sqrt(b)*x + sqrt(b*x
^2 + a)))/b^(7/2)