3.8 \(\int x^3 (A+B x) \left (a+b x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=150 \[ \frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{3 a^3 B x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}-\frac{a \left (a+b x^2\right )^{5/2} (32 A+35 B x)}{560 b^2}+\frac{A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]

[Out]

(3*a^3*B*x*Sqrt[a + b*x^2])/(128*b^2) + (a^2*B*x*(a + b*x^2)^(3/2))/(64*b^2) + (
A*x^2*(a + b*x^2)^(5/2))/(7*b) + (B*x^3*(a + b*x^2)^(5/2))/(8*b) - (a*(32*A + 35
*B*x)*(a + b*x^2)^(5/2))/(560*b^2) + (3*a^4*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2
]])/(128*b^(5/2))

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Rubi [A]  time = 0.327142, 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 \[ \frac{3 a^4 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}+\frac{3 a^3 B x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{3/2}}{64 b^2}-\frac{a \left (a+b x^2\right )^{5/2} (32 A+35 B x)}{560 b^2}+\frac{A x^2 \left (a+b x^2\right )^{5/2}}{7 b}+\frac{B x^3 \left (a+b x^2\right )^{5/2}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(A + B*x)*(a + b*x^2)^(3/2),x]

[Out]

(3*a^3*B*x*Sqrt[a + b*x^2])/(128*b^2) + (a^2*B*x*(a + b*x^2)^(3/2))/(64*b^2) + (
A*x^2*(a + b*x^2)^(5/2))/(7*b) + (B*x^3*(a + b*x^2)^(5/2))/(8*b) - (a*(32*A + 35
*B*x)*(a + b*x^2)^(5/2))/(560*b^2) + (3*a^4*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2
]])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 30.291, size = 139, normalized size = 0.93 \[ \frac{A x^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{7 b} + \frac{3 B a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{5}{2}}} + \frac{3 B a^{3} x \sqrt{a + b x^{2}}}{128 b^{2}} + \frac{B a^{2} x \left (a + b x^{2}\right )^{\frac{3}{2}}}{64 b^{2}} + \frac{B x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}}{8 b} - \frac{a \left (96 A + 105 B x\right ) \left (a + b x^{2}\right )^{\frac{5}{2}}}{1680 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(b*x**2+a)**(3/2),x)

[Out]

A*x**2*(a + b*x**2)**(5/2)/(7*b) + 3*B*a**4*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(1
28*b**(5/2)) + 3*B*a**3*x*sqrt(a + b*x**2)/(128*b**2) + B*a**2*x*(a + b*x**2)**(
3/2)/(64*b**2) + B*x**3*(a + b*x**2)**(5/2)/(8*b) - a*(96*A + 105*B*x)*(a + b*x*
*2)**(5/2)/(1680*b**2)

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Mathematica [A]  time = 0.142776, size = 119, normalized size = 0.79 \[ \frac{105 a^4 B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} \sqrt{a+b x^2} \left (-a^3 (256 A+105 B x)+2 a^2 b x^2 (64 A+35 B x)+8 a b^2 x^4 (128 A+105 B x)+80 b^3 x^6 (8 A+7 B x)\right )}{4480 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(A + B*x)*(a + b*x^2)^(3/2),x]

[Out]

(Sqrt[b]*Sqrt[a + b*x^2]*(80*b^3*x^6*(8*A + 7*B*x) + 2*a^2*b*x^2*(64*A + 35*B*x)
+ 8*a*b^2*x^4*(128*A + 105*B*x) - a^3*(256*A + 105*B*x)) + 105*a^4*B*Log[b*x +
Sqrt[b]*Sqrt[a + b*x^2]])/(4480*b^(5/2))

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Maple [A]  time = 0.012, size = 134, normalized size = 0.9 \[{\frac{A{x}^{2}}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Aa}{35\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bxa}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bx{a}^{2}}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}Bx}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,B{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(b*x^2+a)^(3/2),x)

[Out]

1/7*A*x^2*(b*x^2+a)^(5/2)/b-2/35*A*a/b^2*(b*x^2+a)^(5/2)+1/8*B*x^3*(b*x^2+a)^(5/
2)/b-1/16*B*a/b^2*x*(b*x^2+a)^(5/2)+1/64*B*a^2/b^2*x*(b*x^2+a)^(3/2)+3/128*B*a^3
/b^2*x*(b*x^2+a)^(1/2)+3/128*B*a^4/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)^(3/2)*(B*x + A)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280218, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a^{4} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (560 \, B b^{3} x^{7} + 640 \, A b^{3} x^{6} + 840 \, B a b^{2} x^{5} + 1024 \, A a b^{2} x^{4} + 70 \, B a^{2} b x^{3} + 128 \, A a^{2} b x^{2} - 105 \, B a^{3} x - 256 \, A a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{8960 \, b^{\frac{5}{2}}}, \frac{105 \, B a^{4} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (560 \, B b^{3} x^{7} + 640 \, A b^{3} x^{6} + 840 \, B a b^{2} x^{5} + 1024 \, A a b^{2} x^{4} + 70 \, B a^{2} b x^{3} + 128 \, A a^{2} b x^{2} - 105 \, B a^{3} x - 256 \, A a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{4480 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*(B*x + A)*x^3,x, algorithm="fricas")

[Out]

[1/8960*(105*B*a^4*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(560*
B*b^3*x^7 + 640*A*b^3*x^6 + 840*B*a*b^2*x^5 + 1024*A*a*b^2*x^4 + 70*B*a^2*b*x^3
+ 128*A*a^2*b*x^2 - 105*B*a^3*x - 256*A*a^3)*sqrt(b*x^2 + a)*sqrt(b))/b^(5/2), 1
/4480*(105*B*a^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (560*B*b^3*x^7 + 640*A*b^3
*x^6 + 840*B*a*b^2*x^5 + 1024*A*a*b^2*x^4 + 70*B*a^2*b*x^3 + 128*A*a^2*b*x^2 - 1
05*B*a^3*x - 256*A*a^3)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 24.5491, size = 318, normalized size = 2.12 \[ A a \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + A b \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) - \frac{3 B a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 B a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{B b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(b*x**2+a)**(3/2),x)

[Out]

A*a*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*Piecewise
((8*a**3*sqrt(a + b*x**2)/(105*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)) - 3*B*a**(7/2)*x/(128*b**2*sqrt(1 + b*x**2/a)) - B*a**(5/2)*x**3/(
128*b*sqrt(1 + b*x**2/a)) + 13*B*a**(3/2)*x**5/(64*sqrt(1 + b*x**2/a)) + 5*B*sqr
t(a)*b*x**7/(16*sqrt(1 + b*x**2/a)) + 3*B*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**
(5/2)) + B*b**2*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.228703, size = 155, normalized size = 1.03 \[ -\frac{3 \, B a^{4}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{5}{2}}} - \frac{1}{4480} \, \sqrt{b x^{2} + a}{\left (\frac{256 \, A a^{3}}{b^{2}} +{\left (\frac{105 \, B a^{3}}{b^{2}} - 2 \,{\left (\frac{64 \, A a^{2}}{b} +{\left (\frac{35 \, B a^{2}}{b} + 4 \,{\left (128 \, A a + 5 \,{\left (21 \, B a + 2 \,{\left (7 \, B b x + 8 \, A b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*(B*x + A)*x^3,x, algorithm="giac")

[Out]

-3/128*B*a^4*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2) - 1/4480*sqrt(b*x^2 +
a)*(256*A*a^3/b^2 + (105*B*a^3/b^2 - 2*(64*A*a^2/b + (35*B*a^2/b + 4*(128*A*a +
5*(21*B*a + 2*(7*B*b*x + 8*A*b)*x)*x)*x)*x)*x)*x)