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

Optimal. Leaf size=127 \[ -\frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}-\frac{a^2 A x \sqrt{a+c x^2}}{16 c}-\frac{\left (a+c x^2\right )^{5/2} (12 a B-35 A c x)}{210 c^2}-\frac{a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{B x^2 \left (a+c x^2\right )^{5/2}}{7 c} \]

[Out]

-(a^2*A*x*Sqrt[a + c*x^2])/(16*c) - (a*A*x*(a + c*x^2)^(3/2))/(24*c) + (B*x^2*(a
+ c*x^2)^(5/2))/(7*c) - ((12*a*B - 35*A*c*x)*(a + c*x^2)^(5/2))/(210*c^2) - (a^
3*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(3/2))

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Rubi [A]  time = 0.173697, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^3 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}-\frac{a^2 A x \sqrt{a+c x^2}}{16 c}-\frac{\left (a+c x^2\right )^{5/2} (12 a B-35 A c x)}{210 c^2}-\frac{a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{B x^2 \left (a+c x^2\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*A*x*Sqrt[a + c*x^2])/(16*c) - (a*A*x*(a + c*x^2)^(3/2))/(24*c) + (B*x^2*(a
+ c*x^2)^(5/2))/(7*c) - ((12*a*B - 35*A*c*x)*(a + c*x^2)^(5/2))/(210*c^2) - (a^
3*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(3/2))

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Rubi in Sympy [A]  time = 17.3182, size = 114, normalized size = 0.9 \[ - \frac{A a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 c^{\frac{3}{2}}} - \frac{A a^{2} x \sqrt{a + c x^{2}}}{16 c} - \frac{A a x \left (a + c x^{2}\right )^{\frac{3}{2}}}{24 c} + \frac{B x^{2} \left (a + c x^{2}\right )^{\frac{5}{2}}}{7 c} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (- 35 A c x + 12 B a\right )}{210 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**3*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(16*c**(3/2)) - A*a**2*x*sqrt(a + c*x*
*2)/(16*c) - A*a*x*(a + c*x**2)**(3/2)/(24*c) + B*x**2*(a + c*x**2)**(5/2)/(7*c)
- (a + c*x**2)**(5/2)*(-35*A*c*x + 12*B*a)/(210*c**2)

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Mathematica [A]  time = 0.122067, size = 108, normalized size = 0.85 \[ \frac{\sqrt{a+c x^2} \left (-96 a^3 B+3 a^2 c x (35 A+16 B x)+2 a c^2 x^3 (245 A+192 B x)+40 c^3 x^5 (7 A+6 B x)\right )-105 a^3 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{1680 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(-96*a^3*B + 40*c^3*x^5*(7*A + 6*B*x) + 3*a^2*c*x*(35*A + 16*B*
x) + 2*a*c^2*x^3*(245*A + 192*B*x)) - 105*a^3*A*Sqrt[c]*Log[c*x + Sqrt[c]*Sqrt[a
+ c*x^2]])/(1680*c^2)

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Maple [A]  time = 0.01, size = 113, normalized size = 0.9 \[{\frac{Ax}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aAx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}Ax}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{A{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{B{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Ba}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*A*x*(c*x^2+a)^(5/2)/c-1/24*a*A*x*(c*x^2+a)^(3/2)/c-1/16*a^2*A*x*(c*x^2+a)^(1
/2)/c-1/16*A*a^3/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/7*B*x^2*(c*x^2+a)^(5/2)
/c-2/35*B*a/c^2*(c*x^2+a)^(5/2)

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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((c*x^2 + a)^(3/2)*(B*x + A)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.361697, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A a^{3} c \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 384 \, B a c^{2} x^{4} + 490 \, A a c^{2} x^{3} + 48 \, B a^{2} c x^{2} + 105 \, A a^{2} c x - 96 \, B a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{3360 \, c^{\frac{5}{2}}}, -\frac{105 \, A a^{3} c \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 384 \, B a c^{2} x^{4} + 490 \, A a c^{2} x^{3} + 48 \, B a^{2} c x^{2} + 105 \, A a^{2} c x - 96 \, B a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{1680 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3360*(105*A*a^3*c*log(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(240
*B*c^3*x^6 + 280*A*c^3*x^5 + 384*B*a*c^2*x^4 + 490*A*a*c^2*x^3 + 48*B*a^2*c*x^2
+ 105*A*a^2*c*x - 96*B*a^3)*sqrt(c*x^2 + a)*sqrt(c))/c^(5/2), -1/1680*(105*A*a^3
*c*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (240*B*c^3*x^6 + 280*A*c^3*x^5 + 384*B*a
*c^2*x^4 + 490*A*a*c^2*x^3 + 48*B*a^2*c*x^2 + 105*A*a^2*c*x - 96*B*a^3)*sqrt(c*x
^2 + a)*sqrt(-c))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 32.6354, size = 287, normalized size = 2.26 \[ \frac{A a^{\frac{5}{2}} x}{16 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 A a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{11 A \sqrt{a} c x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{A c^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + B c \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(5/2)*x/(16*c*sqrt(1 + c*x**2/a)) + 17*A*a**(3/2)*x**3/(48*sqrt(1 + c*x**2/
a)) + 11*A*sqrt(a)*c*x**5/(24*sqrt(1 + c*x**2/a)) - A*a**3*asinh(sqrt(c)*x/sqrt(
a))/(16*c**(3/2)) + A*c**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + B*a*Piecewise((
-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(
a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + B*c*Piecewise((8*a**3*sqrt(a
+ c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a
+ c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.274662, size = 139, normalized size = 1.09 \[ \frac{A a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} - \frac{1}{1680} \, \sqrt{c x^{2} + a}{\left (\frac{96 \, B a^{3}}{c^{2}} -{\left (\frac{105 \, A a^{2}}{c} + 2 \,{\left (\frac{24 \, B a^{2}}{c} +{\left (245 \, A a + 4 \,{\left (48 \, B a + 5 \,{\left (6 \, B c x + 7 \, A c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*A*a^3*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2) - 1/1680*sqrt(c*x^2 + a
)*(96*B*a^3/c^2 - (105*A*a^2/c + 2*(24*B*a^2/c + (245*A*a + 4*(48*B*a + 5*(6*B*c
*x + 7*A*c)*x)*x)*x)*x)*x)