3.315 \(\int x^3 (A+B x) \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=127 \[ \frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{a^2 B x \sqrt{a+c x^2}}{16 c^2}-\frac{a \left (a+c x^2\right )^{3/2} (16 A+15 B x)}{120 c^2}+\frac{A x^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{B x^3 \left (a+c x^2\right )^{3/2}}{6 c} \]

[Out]

(a^2*B*x*Sqrt[a + c*x^2])/(16*c^2) + (A*x^2*(a + c*x^2)^(3/2))/(5*c) + (B*x^3*(a
+ c*x^2)^(3/2))/(6*c) - (a*(16*A + 15*B*x)*(a + c*x^2)^(3/2))/(120*c^2) + (a^3*
B*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(5/2))

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Rubi [A]  time = 0.262043, 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 \[ \frac{a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{5/2}}+\frac{a^2 B x \sqrt{a+c x^2}}{16 c^2}-\frac{a \left (a+c x^2\right )^{3/2} (16 A+15 B x)}{120 c^2}+\frac{A x^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{B x^3 \left (a+c x^2\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*B*x*Sqrt[a + c*x^2])/(16*c^2) + (A*x^2*(a + c*x^2)^(3/2))/(5*c) + (B*x^3*(a
+ c*x^2)^(3/2))/(6*c) - (a*(16*A + 15*B*x)*(a + c*x^2)^(3/2))/(120*c^2) + (a^3*
B*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(16*c^(5/2))

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Rubi in Sympy [A]  time = 26.6184, size = 114, normalized size = 0.9 \[ \frac{A x^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}}{5 c} + \frac{B a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 c^{\frac{5}{2}}} + \frac{B a^{2} x \sqrt{a + c x^{2}}}{16 c^{2}} + \frac{B x^{3} \left (a + c x^{2}\right )^{\frac{3}{2}}}{6 c} - \frac{a \left (48 A + 45 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{360 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(c*x**2+a)**(1/2),x)

[Out]

A*x**2*(a + c*x**2)**(3/2)/(5*c) + B*a**3*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(16*
c**(5/2)) + B*a**2*x*sqrt(a + c*x**2)/(16*c**2) + B*x**3*(a + c*x**2)**(3/2)/(6*
c) - a*(48*A + 45*B*x)*(a + c*x**2)**(3/2)/(360*c**2)

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Mathematica [A]  time = 0.104392, size = 100, normalized size = 0.79 \[ \frac{15 a^3 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} \sqrt{a+c x^2} \left (-a^2 (32 A+15 B x)+2 a c x^2 (8 A+5 B x)+8 c^2 x^4 (6 A+5 B x)\right )}{240 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(8*c^2*x^4*(6*A + 5*B*x) + 2*a*c*x^2*(8*A + 5*B*x) - a^
2*(32*A + 15*B*x)) + 15*a^3*B*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(240*c^(5/2))

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Maple [A]  time = 0.01, size = 115, normalized size = 0.9 \[{\frac{A{x}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,aA}{15\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{x}^{3}}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aBx}{8\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}Bx}{16\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{B{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+a)^(1/2),x)

[Out]

1/5*A*x^2*(c*x^2+a)^(3/2)/c-2/15*A*a/c^2*(c*x^2+a)^(3/2)+1/6*B*x^3*(c*x^2+a)^(3/
2)/c-1/8*B*a/c^2*x*(c*x^2+a)^(3/2)+1/16*a^2*B*x*(c*x^2+a)^(1/2)/c^2+1/16*B*a^3/c
^(5/2)*ln(c^(1/2)*x+(c*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(sqrt(c*x^2 + a)*(B*x + A)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306353, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a^{3} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 10 \, B a c x^{3} + 16 \, A a c x^{2} - 15 \, B a^{2} x - 32 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{480 \, c^{\frac{5}{2}}}, \frac{15 \, B a^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 10 \, B a c x^{3} + 16 \, A a c x^{2} - 15 \, B a^{2} x - 32 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{240 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/480*(15*B*a^3*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(40*B*c
^2*x^5 + 48*A*c^2*x^4 + 10*B*a*c*x^3 + 16*A*a*c*x^2 - 15*B*a^2*x - 32*A*a^2)*sqr
t(c*x^2 + a)*sqrt(c))/c^(5/2), 1/240*(15*B*a^3*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)
) + (40*B*c^2*x^5 + 48*A*c^2*x^4 + 10*B*a*c*x^3 + 16*A*a*c*x^2 - 15*B*a^2*x - 32
*A*a^2)*sqrt(c*x^2 + a)*sqrt(-c))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 20.572, size = 192, normalized size = 1.51 \[ A \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) - \frac{B a^{\frac{5}{2}} x}{16 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{\frac{3}{2}} x^{3}}{48 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 B \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{5}{2}}} + \frac{B c x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(c*x**2+a)**(1/2),x)

[Out]

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*a**(5/2)*x/(1
6*c**2*sqrt(1 + c*x**2/a)) - B*a**(3/2)*x**3/(48*c*sqrt(1 + c*x**2/a)) + 5*B*sqr
t(a)*x**5/(24*sqrt(1 + c*x**2/a)) + B*a**3*asinh(sqrt(c)*x/sqrt(a))/(16*c**(5/2)
) + B*c*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.275102, size = 126, normalized size = 0.99 \[ -\frac{B a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{5}{2}}} + \frac{1}{240} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, B x + 6 \, A\right )} x + \frac{5 \, B a}{c}\right )} x + \frac{8 \, A a}{c}\right )} x - \frac{15 \, B a^{2}}{c^{2}}\right )} x - \frac{32 \, A a^{2}}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*B*a^3*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/240*sqrt(c*x^2 + a
)*((2*((4*(5*B*x + 6*A)*x + 5*B*a/c)*x + 8*A*a/c)*x - 15*B*a^2/c^2)*x - 32*A*a^2
/c^2)