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

Optimal. Leaf size=126 \[ -\frac{5 a^4 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}-\frac{5 a^3 B x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{\left (a+b x^2\right )^{7/2} (8 A+7 B x)}{56 b}-\frac{a B x \left (a+b x^2\right )^{5/2}}{48 b} \]

[Out]

(-5*a^3*B*x*Sqrt[a + b*x^2])/(128*b) - (5*a^2*B*x*(a + b*x^2)^(3/2))/(192*b) - (
a*B*x*(a + b*x^2)^(5/2))/(48*b) + ((8*A + 7*B*x)*(a + b*x^2)^(7/2))/(56*b) - (5*
a^4*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(3/2))

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Rubi [A]  time = 0.131859, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{5 a^4 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}-\frac{5 a^3 B x \sqrt{a+b x^2}}{128 b}-\frac{5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}+\frac{\left (a+b x^2\right )^{7/2} (8 A+7 B x)}{56 b}-\frac{a B x \left (a+b x^2\right )^{5/2}}{48 b} \]

Antiderivative was successfully verified.

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

[Out]

(-5*a^3*B*x*Sqrt[a + b*x^2])/(128*b) - (5*a^2*B*x*(a + b*x^2)^(3/2))/(192*b) - (
a*B*x*(a + b*x^2)^(5/2))/(48*b) + ((8*A + 7*B*x)*(a + b*x^2)^(7/2))/(56*b) - (5*
a^4*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(3/2))

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Rubi in Sympy [A]  time = 13.4098, size = 116, normalized size = 0.92 \[ - \frac{5 B a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{3}{2}}} - \frac{5 B a^{3} x \sqrt{a + b x^{2}}}{128 b} - \frac{5 B a^{2} x \left (a + b x^{2}\right )^{\frac{3}{2}}}{192 b} - \frac{B a x \left (a + b x^{2}\right )^{\frac{5}{2}}}{48 b} + \frac{\left (8 A + 7 B x\right ) \left (a + b x^{2}\right )^{\frac{7}{2}}}{56 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*B*a**4*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(128*b**(3/2)) - 5*B*a**3*x*sqrt(a +
b*x**2)/(128*b) - 5*B*a**2*x*(a + b*x**2)**(3/2)/(192*b) - B*a*x*(a + b*x**2)**
(5/2)/(48*b) + (8*A + 7*B*x)*(a + b*x**2)**(7/2)/(56*b)

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Mathematica [A]  time = 0.146143, size = 119, normalized size = 0.94 \[ \frac{\sqrt{b} \sqrt{a+b x^2} \left (3 a^3 (128 A+35 B x)+2 a^2 b x^2 (576 A+413 B x)+8 a b^2 x^4 (144 A+119 B x)+48 b^3 x^6 (8 A+7 B x)\right )-105 a^4 B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2688 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[a + b*x^2]*(48*b^3*x^6*(8*A + 7*B*x) + 3*a^3*(128*A + 35*B*x) + 8*
a*b^2*x^4*(144*A + 119*B*x) + 2*a^2*b*x^2*(576*A + 413*B*x)) - 105*a^4*B*Log[b*x
+ Sqrt[b]*Sqrt[a + b*x^2]])/(2688*b^(3/2))

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Maple [A]  time = 0.007, size = 113, normalized size = 0.9 \[{\frac{A}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bxa}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Bx{a}^{2}}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}Bx}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267743, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a^{4} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (336 \, B b^{3} x^{7} + 384 \, A b^{3} x^{6} + 952 \, B a b^{2} x^{5} + 1152 \, A a b^{2} x^{4} + 826 \, B a^{2} b x^{3} + 1152 \, A a^{2} b x^{2} + 105 \, B a^{3} x + 384 \, A a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{5376 \, b^{\frac{3}{2}}}, -\frac{105 \, B a^{4} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (336 \, B b^{3} x^{7} + 384 \, A b^{3} x^{6} + 952 \, B a b^{2} x^{5} + 1152 \, A a b^{2} x^{4} + 826 \, B a^{2} b x^{3} + 1152 \, A a^{2} b x^{2} + 105 \, B a^{3} x + 384 \, A a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{2688 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/5376*(105*B*a^4*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(336*B
*b^3*x^7 + 384*A*b^3*x^6 + 952*B*a*b^2*x^5 + 1152*A*a*b^2*x^4 + 826*B*a^2*b*x^3
+ 1152*A*a^2*b*x^2 + 105*B*a^3*x + 384*A*a^3)*sqrt(b*x^2 + a)*sqrt(b))/b^(3/2),
-1/2688*(105*B*a^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (336*B*b^3*x^7 + 384*A*b
^3*x^6 + 952*B*a*b^2*x^5 + 1152*A*a*b^2*x^4 + 826*B*a^2*b*x^3 + 1152*A*a^2*b*x^2
+ 105*B*a^3*x + 384*A*a^3)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*b)]

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Sympy [A]  time = 30.001, size = 354, normalized size = 2.81 \[ A a^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + 2 A a b \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + A b^{2} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{5 B a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 B a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 B a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 B \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{B b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*Piecewise((sqrt(a)*x**2/2, Eq(b, 0)), ((a + b*x**2)**(3/2)/(3*b), True))
+ 2*A*a*b*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)) + A*b**2*
Piecewise((8*a**3*sqrt(a + b*x**2)/(105*b**3) - 4*a**2*x**2*sqrt(a + b*x**2)/(10
5*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)) + 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/s
qrt(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 = 0.230988, size = 154, normalized size = 1.22 \[ \frac{5 \, B a^{4}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} + \frac{1}{2688} \,{\left (\frac{384 \, A a^{3}}{b} +{\left (\frac{105 \, B a^{3}}{b} + 2 \,{\left (576 \, A a^{2} +{\left (413 \, B a^{2} + 4 \,{\left (144 \, A a b +{\left (119 \, B a b + 6 \,{\left (7 \, B b^{2} x + 8 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{b x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/128*B*a^4*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + 1/2688*(384*A*a^3/b
+ (105*B*a^3/b + 2*(576*A*a^2 + (413*B*a^2 + 4*(144*A*a*b + (119*B*a*b + 6*(7*B*
b^2*x + 8*A*b^2)*x)*x)*x)*x)*x)*x)*sqrt(b*x^2 + a)