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

Optimal. Leaf size=149 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 A b-a B)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 A b-a B)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 A b-a B)}{192 b}+\frac{B x \left (a+b x^2\right )^{7/2}}{8 b} \]

[Out]

(5*a^2*(8*A*b - a*B)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*A*b - a*B)*x*(a + b*x^
2)^(3/2))/(192*b) + ((8*A*b - a*B)*x*(a + b*x^2)^(5/2))/(48*b) + (B*x*(a + b*x^2
)^(7/2))/(8*b) + (5*a^3*(8*A*b - a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128
*b^(3/2))

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Rubi [A]  time = 0.137666, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 A b-a B)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 A b-a B)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 A b-a B)}{192 b}+\frac{B x \left (a+b x^2\right )^{7/2}}{8 b} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^2*(8*A*b - a*B)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*A*b - a*B)*x*(a + b*x^
2)^(3/2))/(192*b) + ((8*A*b - a*B)*x*(a + b*x^2)^(5/2))/(48*b) + (B*x*(a + b*x^2
)^(7/2))/(8*b) + (5*a^3*(8*A*b - a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128
*b^(3/2))

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Rubi in Sympy [A]  time = 14.5118, size = 134, normalized size = 0.9 \[ \frac{B x \left (a + b x^{2}\right )^{\frac{7}{2}}}{8 b} + \frac{5 a^{3} \left (8 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{3}{2}}} + \frac{5 a^{2} x \sqrt{a + b x^{2}} \left (8 A b - B a\right )}{128 b} + \frac{5 a x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (8 A b - B a\right )}{192 b} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (8 A b - B a\right )}{48 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x*(a + b*x**2)**(7/2)/(8*b) + 5*a**3*(8*A*b - B*a)*atanh(sqrt(b)*x/sqrt(a + b*
x**2))/(128*b**(3/2)) + 5*a**2*x*sqrt(a + b*x**2)*(8*A*b - B*a)/(128*b) + 5*a*x*
(a + b*x**2)**(3/2)*(8*A*b - B*a)/(192*b) + x*(a + b*x**2)**(5/2)*(8*A*b - B*a)/
(48*b)

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Mathematica [A]  time = 0.121125, size = 121, normalized size = 0.81 \[ \sqrt{a+b x^2} \left (\frac{a^2 x (5 a B+88 A b)}{128 b}+\frac{1}{48} b x^5 (17 a B+8 A b)+\frac{1}{192} a x^3 (59 a B+104 A b)+\frac{1}{8} b^2 B x^7\right )-\frac{5 a^3 (a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a^2*(88*A*b + 5*a*B)*x)/(128*b) + (a*(104*A*b + 59*a*B)*x^3)/1
92 + (b*(8*A*b + 17*a*B)*x^5)/48 + (b^2*B*x^7)/8) - (5*a^3*(-8*A*b + a*B)*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]])/(128*b^(3/2))

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Maple [A]  time = 0.008, size = 166, normalized size = 1.1 \[{\frac{Ax}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aAx}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Ax}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,A{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{Bx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bxa}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Bx{a}^{2}}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*A*x*(b*x^2+a)^(5/2)+5/24*A*a*x*(b*x^2+a)^(3/2)+5/16*A*a^2*x*(b*x^2+a)^(1/2)+
5/16*A*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/8*B*x*(b*x^2+a)^(7/2)/b-1/48*
B*a/b*x*(b*x^2+a)^(5/2)-5/192*B*a^2/b*x*(b*x^2+a)^(3/2)-5/128*B*a^3/b*x*(b*x^2+a
)^(1/2)-5/128*B*a^4/b^(3/2)*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308742, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{7} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{5} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{768 \, b^{\frac{3}{2}}}, \frac{{\left (48 \, B b^{3} x^{7} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{5} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{384 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(2*(48*B*b^3*x^7 + 8*(17*B*a*b^2 + 8*A*b^3)*x^5 + 2*(59*B*a^2*b + 104*A*a
*b^2)*x^3 + 3*(5*B*a^3 + 88*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(B*a^4 - 8*
A*a^3*b)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(3/2), 1/384*((4
8*B*b^3*x^7 + 8*(17*B*a*b^2 + 8*A*b^3)*x^5 + 2*(59*B*a^2*b + 104*A*a*b^2)*x^3 +
3*(5*B*a^3 + 88*A*a^2*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 15*(B*a^4 - 8*A*a^3*b)*ar
ctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b)]

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Sympy [A]  time = 87.4629, size = 316, normalized size = 2.12 \[ \frac{A a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 A a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 A a^{\frac{3}{2}} b x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 A \sqrt{a} b^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{A b^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 B a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 B a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 B \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{B b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(5/2)*x*sqrt(1 + b*x**2/a)/2 + 3*A*a**(5/2)*x/(16*sqrt(1 + b*x**2/a)) + 35*
A*a**(3/2)*b*x**3/(48*sqrt(1 + b*x**2/a)) + 17*A*sqrt(a)*b**2*x**5/(24*sqrt(1 +
b*x**2/a)) + 5*A*a**3*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b)) + A*b**3*x**7/(6*sqr
t(a)*sqrt(1 + b*x**2/a)) + 5*B*a**(7/2)*x/(128*b*sqrt(1 + b*x**2/a)) + 133*B*a**
(5/2)*x**3/(384*sqrt(1 + b*x**2/a)) + 127*B*a**(3/2)*b*x**5/(192*sqrt(1 + b*x**2
/a)) + 23*B*sqrt(a)*b**2*x**7/(48*sqrt(1 + b*x**2/a)) - 5*B*a**4*asinh(sqrt(b)*x
/sqrt(a))/(128*b**(3/2)) + B*b**3*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.249302, size = 181, normalized size = 1.21 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, B b^{2} x^{2} + \frac{17 \, B a b^{7} + 8 \, A b^{8}}{b^{6}}\right )} x^{2} + \frac{59 \, B a^{2} b^{6} + 104 \, A a b^{7}}{b^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, B a^{3} b^{5} + 88 \, A a^{2} b^{6}\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x + \frac{5 \,{\left (B a^{4} - 8 \, A a^{3} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(2*(4*(6*B*b^2*x^2 + (17*B*a*b^7 + 8*A*b^8)/b^6)*x^2 + (59*B*a^2*b^6 + 104
*A*a*b^7)/b^6)*x^2 + 3*(5*B*a^3*b^5 + 88*A*a^2*b^6)/b^6)*sqrt(b*x^2 + a)*x + 5/1
28*(B*a^4 - 8*A*a^3*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2)