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

Optimal. Leaf size=150 \[ -\frac{5 a^4 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}-\frac{5 a^3 A x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 A x \left (a+b x^2\right )^{3/2}}{192 b}-\frac{\left (a+b x^2\right )^{7/2} (16 a B-63 A b x)}{504 b^2}-\frac{a A x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{B x^2 \left (a+b x^2\right )^{7/2}}{9 b} \]

[Out]

(-5*a^3*A*x*Sqrt[a + b*x^2])/(128*b) - (5*a^2*A*x*(a + b*x^2)^(3/2))/(192*b) - (
a*A*x*(a + b*x^2)^(5/2))/(48*b) + (B*x^2*(a + b*x^2)^(7/2))/(9*b) - ((16*a*B - 6
3*A*b*x)*(a + b*x^2)^(7/2))/(504*b^2) - (5*a^4*A*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*
x^2]])/(128*b^(3/2))

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Rubi [A]  time = 0.22998, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{5 a^4 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}-\frac{5 a^3 A x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 A x \left (a+b x^2\right )^{3/2}}{192 b}-\frac{\left (a+b x^2\right )^{7/2} (16 a B-63 A b x)}{504 b^2}-\frac{a A x \left (a+b x^2\right )^{5/2}}{48 b}+\frac{B x^2 \left (a+b x^2\right )^{7/2}}{9 b} \]

Antiderivative was successfully verified.

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

[Out]

(-5*a^3*A*x*Sqrt[a + b*x^2])/(128*b) - (5*a^2*A*x*(a + b*x^2)^(3/2))/(192*b) - (
a*A*x*(a + b*x^2)^(5/2))/(48*b) + (B*x^2*(a + b*x^2)^(7/2))/(9*b) - ((16*a*B - 6
3*A*b*x)*(a + b*x^2)^(7/2))/(504*b^2) - (5*a^4*A*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*
x^2]])/(128*b^(3/2))

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Rubi in Sympy [A]  time = 20.6821, size = 139, normalized size = 0.93 \[ - \frac{5 A a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{3}{2}}} - \frac{5 A a^{3} x \sqrt{a + b x^{2}}}{128 b} - \frac{5 A a^{2} x \left (a + b x^{2}\right )^{\frac{3}{2}}}{192 b} - \frac{A a x \left (a + b x^{2}\right )^{\frac{5}{2}}}{48 b} + \frac{B x^{2} \left (a + b x^{2}\right )^{\frac{7}{2}}}{9 b} - \frac{\left (a + b x^{2}\right )^{\frac{7}{2}} \left (- 63 A b x + 16 B a\right )}{504 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*A*a**4*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(128*b**(3/2)) - 5*A*a**3*x*sqrt(a +
b*x**2)/(128*b) - 5*A*a**2*x*(a + b*x**2)**(3/2)/(192*b) - A*a*x*(a + b*x**2)**
(5/2)/(48*b) + B*x**2*(a + b*x**2)**(7/2)/(9*b) - (a + b*x**2)**(7/2)*(-63*A*b*x
+ 16*B*a)/(504*b**2)

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Mathematica [A]  time = 0.153675, size = 126, normalized size = 0.84 \[ \frac{\sqrt{a+b x^2} \left (-256 a^4 B+a^3 b x (315 A+128 B x)+6 a^2 b^2 x^3 (413 A+320 B x)+8 a b^3 x^5 (357 A+304 B x)+112 b^4 x^7 (9 A+8 B x)\right )-315 a^4 A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8064 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*(-256*a^4*B + 112*b^4*x^7*(9*A + 8*B*x) + a^3*b*x*(315*A + 128*
B*x) + 8*a*b^3*x^5*(357*A + 304*B*x) + 6*a^2*b^2*x^3*(413*A + 320*B*x)) - 315*a^
4*A*Sqrt[b]*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8064*b^2)

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Maple [A]  time = 0.008, size = 132, normalized size = 0.9 \[{\frac{Ax}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{aAx}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}Ax}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}Ax}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,A{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{2}}{9\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Ba}{63\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*A*x*(b*x^2+a)^(7/2)/b-1/48*a*A*x*(b*x^2+a)^(5/2)/b-5/192*a^2*A*x*(b*x^2+a)^(
3/2)/b-5/128*a^3*A*x*(b*x^2+a)^(1/2)/b-5/128*A*a^4/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a
)^(1/2))+1/9*B*x^2*(b*x^2+a)^(7/2)/b-2/63*B*a/b^2*(b*x^2+a)^(7/2)

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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)^(5/2)*(B*x + A)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270992, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, A a^{4} b \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (896 \, B b^{4} x^{8} + 1008 \, A b^{4} x^{7} + 2432 \, B a b^{3} x^{6} + 2856 \, A a b^{3} x^{5} + 1920 \, B a^{2} b^{2} x^{4} + 2478 \, A a^{2} b^{2} x^{3} + 128 \, B a^{3} b x^{2} + 315 \, A a^{3} b x - 256 \, B a^{4}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{16128 \, b^{\frac{5}{2}}}, -\frac{315 \, A a^{4} b \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (896 \, B b^{4} x^{8} + 1008 \, A b^{4} x^{7} + 2432 \, B a b^{3} x^{6} + 2856 \, A a b^{3} x^{5} + 1920 \, B a^{2} b^{2} x^{4} + 2478 \, A a^{2} b^{2} x^{3} + 128 \, B a^{3} b x^{2} + 315 \, A a^{3} b x - 256 \, B a^{4}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{8064 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16128*(315*A*a^4*b*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(89
6*B*b^4*x^8 + 1008*A*b^4*x^7 + 2432*B*a*b^3*x^6 + 2856*A*a*b^3*x^5 + 1920*B*a^2*
b^2*x^4 + 2478*A*a^2*b^2*x^3 + 128*B*a^3*b*x^2 + 315*A*a^3*b*x - 256*B*a^4)*sqrt
(b*x^2 + a)*sqrt(b))/b^(5/2), -1/8064*(315*A*a^4*b*arctan(sqrt(-b)*x/sqrt(b*x^2
+ a)) - (896*B*b^4*x^8 + 1008*A*b^4*x^7 + 2432*B*a*b^3*x^6 + 2856*A*a*b^3*x^5 +
1920*B*a^2*b^2*x^4 + 2478*A*a^2*b^2*x^3 + 128*B*a^3*b*x^2 + 315*A*a^3*b*x - 256*
B*a^4)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 32.2609, size = 442, normalized size = 2.95 \[ \frac{5 A a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 A a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 A a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 A \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 A a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{A b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + B a^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 B a b \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + B b^{2} \left (\begin{cases} - \frac{16 a^{4} \sqrt{a + b x^{2}}}{315 b^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{6} \sqrt{a + b x^{2}}}{63 b} + \frac{x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x+A)*(b*x**2+a)**(5/2),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)) + B*a**2*Piecewise((-2*a**2*sqrt(a +
b*x**2)/(15*b**2) + a*x**2*sqrt(a + b*x**2)/(15*b) + x**4*sqrt(a + b*x**2)/5, Ne
(b, 0)), (sqrt(a)*x**4/4, True)) + 2*B*a*b*Piecewise((8*a**3*sqrt(a + b*x**2)/(1
05*b**3) - 4*a**2*x**2*sqrt(a + b*x**2)/(105*b**2) + a*x**4*sqrt(a + b*x**2)/(35
*b) + x**6*sqrt(a + b*x**2)/7, Ne(b, 0)), (sqrt(a)*x**6/6, True)) + B*b**2*Piece
wise((-16*a**4*sqrt(a + b*x**2)/(315*b**4) + 8*a**3*x**2*sqrt(a + b*x**2)/(315*b
**3) - 2*a**2*x**4*sqrt(a + b*x**2)/(105*b**2) + a*x**6*sqrt(a + b*x**2)/(63*b)
+ x**8*sqrt(a + b*x**2)/9, Ne(b, 0)), (sqrt(a)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.224881, size = 173, normalized size = 1.15 \[ \frac{5 \, A a^{4}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} - \frac{1}{8064} \,{\left (\frac{256 \, B a^{4}}{b^{2}} -{\left (\frac{315 \, A a^{3}}{b} + 2 \,{\left (\frac{64 \, B a^{3}}{b} +{\left (1239 \, A a^{2} + 4 \,{\left (240 \, B a^{2} +{\left (357 \, A a b + 2 \,{\left (152 \, B a b + 7 \,{\left (8 \, B b^{2} x + 9 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{b x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/128*A*a^4*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) - 1/8064*(256*B*a^4/b^
2 - (315*A*a^3/b + 2*(64*B*a^3/b + (1239*A*a^2 + 4*(240*B*a^2 + (357*A*a*b + 2*(
152*B*a*b + 7*(8*B*b^2*x + 9*A*b^2)*x)*x)*x)*x)*x)*x)*x)*sqrt(b*x^2 + a)