3.15 \(\int x^3 (A+B x) \left (a+b x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=173 \[ \frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}-\frac{a \left (a+b x^2\right )^{7/2} (160 A+189 B x)}{5040 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]

[Out]

(3*a^4*B*x*Sqrt[a + b*x^2])/(256*b^2) + (a^3*B*x*(a + b*x^2)^(3/2))/(128*b^2) +
(a^2*B*x*(a + b*x^2)^(5/2))/(160*b^2) + (A*x^2*(a + b*x^2)^(7/2))/(9*b) + (B*x^3
*(a + b*x^2)^(7/2))/(10*b) - (a*(160*A + 189*B*x)*(a + b*x^2)^(7/2))/(5040*b^2)
+ (3*a^5*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))

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Rubi [A]  time = 0.356346, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}-\frac{a \left (a+b x^2\right )^{7/2} (160 A+189 B x)}{5040 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(A + B*x)*(a + b*x^2)^(5/2),x]

[Out]

(3*a^4*B*x*Sqrt[a + b*x^2])/(256*b^2) + (a^3*B*x*(a + b*x^2)^(3/2))/(128*b^2) +
(a^2*B*x*(a + b*x^2)^(5/2))/(160*b^2) + (A*x^2*(a + b*x^2)^(7/2))/(9*b) + (B*x^3
*(a + b*x^2)^(7/2))/(10*b) - (a*(160*A + 189*B*x)*(a + b*x^2)^(7/2))/(5040*b^2)
+ (3*a^5*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))

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Rubi in Sympy [A]  time = 34.7748, size = 162, normalized size = 0.94 \[ \frac{A x^{2} \left (a + b x^{2}\right )^{\frac{7}{2}}}{9 b} + \frac{3 B a^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{5}{2}}} + \frac{3 B a^{4} x \sqrt{a + b x^{2}}}{256 b^{2}} + \frac{B a^{3} x \left (a + b x^{2}\right )^{\frac{3}{2}}}{128 b^{2}} + \frac{B a^{2} x \left (a + b x^{2}\right )^{\frac{5}{2}}}{160 b^{2}} + \frac{B x^{3} \left (a + b x^{2}\right )^{\frac{7}{2}}}{10 b} - \frac{a \left (160 A + 189 B x\right ) \left (a + b x^{2}\right )^{\frac{7}{2}}}{5040 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(b*x**2+a)**(5/2),x)

[Out]

A*x**2*(a + b*x**2)**(7/2)/(9*b) + 3*B*a**5*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2
56*b**(5/2)) + 3*B*a**4*x*sqrt(a + b*x**2)/(256*b**2) + B*a**3*x*(a + b*x**2)**(
3/2)/(128*b**2) + B*a**2*x*(a + b*x**2)**(5/2)/(160*b**2) + B*x**3*(a + b*x**2)*
*(7/2)/(10*b) - a*(160*A + 189*B*x)*(a + b*x**2)**(7/2)/(5040*b**2)

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Mathematica [A]  time = 0.176267, size = 138, normalized size = 0.8 \[ \frac{945 a^5 B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} \sqrt{a+b x^2} \left (-5 a^4 (512 A+189 B x)+10 a^3 b x^2 (128 A+63 B x)+24 a^2 b^2 x^4 (800 A+651 B x)+16 a b^3 x^6 (1520 A+1323 B x)+896 b^4 x^8 (10 A+9 B x)\right )}{80640 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(A + B*x)*(a + b*x^2)^(5/2),x]

[Out]

(Sqrt[b]*Sqrt[a + b*x^2]*(896*b^4*x^8*(10*A + 9*B*x) + 10*a^3*b*x^2*(128*A + 63*
B*x) - 5*a^4*(512*A + 189*B*x) + 24*a^2*b^2*x^4*(800*A + 651*B*x) + 16*a*b^3*x^6
*(1520*A + 1323*B*x)) + 945*a^5*B*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(80640*b^(
5/2))

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Maple [A]  time = 0.012, size = 153, normalized size = 0.9 \[{\frac{A{x}^{2}}{9\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Aa}{63\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Bxa}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bx{a}^{2}}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{3}Bx}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{4}Bx}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,B{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(b*x^2+a)^(5/2),x)

[Out]

1/9*A*x^2*(b*x^2+a)^(7/2)/b-2/63*A*a/b^2*(b*x^2+a)^(7/2)+1/10*B*x^3*(b*x^2+a)^(7
/2)/b-3/80*B*a/b^2*x*(b*x^2+a)^(7/2)+1/160*a^2*B*x*(b*x^2+a)^(5/2)/b^2+1/128*a^3
*B*x*(b*x^2+a)^(3/2)/b^2+3/256*a^4*B*x*(b*x^2+a)^(1/2)/b^2+3/256*B*a^5/b^(5/2)*l
n(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282127, size = 1, normalized size = 0.01 \[ \left [\frac{945 \, B a^{5} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (8064 \, B b^{4} x^{9} + 8960 \, A b^{4} x^{8} + 21168 \, B a b^{3} x^{7} + 24320 \, A a b^{3} x^{6} + 15624 \, B a^{2} b^{2} x^{5} + 19200 \, A a^{2} b^{2} x^{4} + 630 \, B a^{3} b x^{3} + 1280 \, A a^{3} b x^{2} - 945 \, B a^{4} x - 2560 \, A a^{4}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{161280 \, b^{\frac{5}{2}}}, \frac{945 \, B a^{5} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8064 \, B b^{4} x^{9} + 8960 \, A b^{4} x^{8} + 21168 \, B a b^{3} x^{7} + 24320 \, A a b^{3} x^{6} + 15624 \, B a^{2} b^{2} x^{5} + 19200 \, A a^{2} b^{2} x^{4} + 630 \, B a^{3} b x^{3} + 1280 \, A a^{3} b x^{2} - 945 \, B a^{4} x - 2560 \, A a^{4}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{80640 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*(B*x + A)*x^3,x, algorithm="fricas")

[Out]

[1/161280*(945*B*a^5*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(80
64*B*b^4*x^9 + 8960*A*b^4*x^8 + 21168*B*a*b^3*x^7 + 24320*A*a*b^3*x^6 + 15624*B*
a^2*b^2*x^5 + 19200*A*a^2*b^2*x^4 + 630*B*a^3*b*x^3 + 1280*A*a^3*b*x^2 - 945*B*a
^4*x - 2560*A*a^4)*sqrt(b*x^2 + a)*sqrt(b))/b^(5/2), 1/80640*(945*B*a^5*arctan(s
qrt(-b)*x/sqrt(b*x^2 + a)) + (8064*B*b^4*x^9 + 8960*A*b^4*x^8 + 21168*B*a*b^3*x^
7 + 24320*A*a*b^3*x^6 + 15624*B*a^2*b^2*x^5 + 19200*A*a^2*b^2*x^4 + 630*B*a^3*b*
x^3 + 1280*A*a^3*b*x^2 - 945*B*a^4*x - 2560*A*a^4)*sqrt(b*x^2 + a)*sqrt(-b))/(sq
rt(-b)*b^2)]

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Sympy [A]  time = 46.6717, size = 469, normalized size = 2.71 \[ A a^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 A a b \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + A b^{2} \left (\begin{cases} - \frac{16 a^{4} \sqrt{a + b x^{2}}}{315 b^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{6} \sqrt{a + b x^{2}}}{63 b} + \frac{x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) - \frac{3 B a^{\frac{9}{2}} x}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{7}{2}} x^{3}}{256 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{129 B a^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{73 B a^{\frac{3}{2}} b x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 B \sqrt{a} b^{2} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{5}{2}}} + \frac{B b^{3} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(b*x**2+a)**(5/2),x)

[Out]

A*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*A*a*b*Pi
ecewise((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)), (sq
rt(a)*x**6/6, True)) + A*b**2*Piecewise((-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)) - 3*B*a**(9/2)*x/(256*b**2*sqrt(1 + b*x**2/a)) - B*a**(7/2)*x**
3/(256*b*sqrt(1 + b*x**2/a)) + 129*B*a**(5/2)*x**5/(640*sqrt(1 + b*x**2/a)) + 73
*B*a**(3/2)*b*x**7/(160*sqrt(1 + b*x**2/a)) + 29*B*sqrt(a)*b**2*x**9/(80*sqrt(1
+ b*x**2/a)) + 3*B*a**5*asinh(sqrt(b)*x/sqrt(a))/(256*b**(5/2)) + B*b**3*x**11/(
10*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.228995, size = 189, normalized size = 1.09 \[ -\frac{3 \, B a^{5}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} - \frac{1}{80640} \,{\left (\frac{2560 \, A a^{4}}{b^{2}} +{\left (\frac{945 \, B a^{4}}{b^{2}} - 2 \,{\left (\frac{640 \, A a^{3}}{b} +{\left (\frac{315 \, B a^{3}}{b} + 4 \,{\left (2400 \, A a^{2} +{\left (1953 \, B a^{2} + 2 \,{\left (1520 \, A a b + 7 \,{\left (189 \, B a b + 8 \,{\left (9 \, B b^{2} x + 10 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{b x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*(B*x + A)*x^3,x, algorithm="giac")

[Out]

-3/256*B*a^5*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2) - 1/80640*(2560*A*a^4
/b^2 + (945*B*a^4/b^2 - 2*(640*A*a^3/b + (315*B*a^3/b + 4*(2400*A*a^2 + (1953*B*
a^2 + 2*(1520*A*a*b + 7*(189*B*a*b + 8*(9*B*b^2*x + 10*A*b^2)*x)*x)*x)*x)*x)*x)*
x)*x)*sqrt(b*x^2 + a)