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

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

[Out]

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

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Rubi [A]  time = 0.302416, 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 \[ \frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{3 a^4 B x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac{a \left (a+c x^2\right )^{7/2} (160 A+189 B x)}{5040 c^2}+\frac{A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{B x^3 \left (a+c x^2\right )^{7/2}}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.7364, size = 162, normalized size = 0.94 \[ \frac{A x^{2} \left (a + c x^{2}\right )^{\frac{7}{2}}}{9 c} + \frac{3 B a^{5} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{256 c^{\frac{5}{2}}} + \frac{3 B a^{4} x \sqrt{a + c x^{2}}}{256 c^{2}} + \frac{B a^{3} x \left (a + c x^{2}\right )^{\frac{3}{2}}}{128 c^{2}} + \frac{B a^{2} x \left (a + c x^{2}\right )^{\frac{5}{2}}}{160 c^{2}} + \frac{B x^{3} \left (a + c x^{2}\right )^{\frac{7}{2}}}{10 c} - \frac{a \left (160 A + 189 B x\right ) \left (a + c x^{2}\right )^{\frac{7}{2}}}{5040 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.209337, size = 138, normalized size = 0.8 \[ \frac{945 a^5 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} \sqrt{a+c x^2} \left (-5 a^4 (512 A+189 B x)+10 a^3 c x^2 (128 A+63 B x)+24 a^2 c^2 x^4 (800 A+651 B x)+16 a c^3 x^6 (1520 A+1323 B x)+896 c^4 x^8 (10 A+9 B x)\right )}{80640 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 153, normalized size = 0.9 \[{\frac{A{x}^{2}}{9\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,aA}{63\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{x}^{3}}{10\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,aBx}{80\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}Bx}{160\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{3}Bx}{128\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{4}Bx}{256\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,B{a}^{5}}{256}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.34215, size = 1, normalized size = 0.01 \[ \left [\frac{945 \, B a^{5} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8064 \, B c^{4} x^{9} + 8960 \, A c^{4} x^{8} + 21168 \, B a c^{3} x^{7} + 24320 \, A a c^{3} x^{6} + 15624 \, B a^{2} c^{2} x^{5} + 19200 \, A a^{2} c^{2} x^{4} + 630 \, B a^{3} c x^{3} + 1280 \, A a^{3} c x^{2} - 945 \, B a^{4} x - 2560 \, A a^{4}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{161280 \, c^{\frac{5}{2}}}, \frac{945 \, B a^{5} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (8064 \, B c^{4} x^{9} + 8960 \, A c^{4} x^{8} + 21168 \, B a c^{3} x^{7} + 24320 \, A a c^{3} x^{6} + 15624 \, B a^{2} c^{2} x^{5} + 19200 \, A a^{2} c^{2} x^{4} + 630 \, B a^{3} c x^{3} + 1280 \, A a^{3} c x^{2} - 945 \, B a^{4} x - 2560 \, A a^{4}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{80640 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 91.5573, size = 469, normalized size = 2.71 \[ A a^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 A a c \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + A c^{2} \left (\begin{cases} - \frac{16 a^{4} \sqrt{a + c x^{2}}}{315 c^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + c x^{2}}}{315 c^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{6} \sqrt{a + c x^{2}}}{63 c} + \frac{x^{8} \sqrt{a + c x^{2}}}{9} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) - \frac{3 B a^{\frac{9}{2}} x}{256 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{\frac{7}{2}} x^{3}}{256 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{129 B a^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{73 B a^{\frac{3}{2}} c x^{7}}{160 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{29 B \sqrt{a} c^{2} x^{9}}{80 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B a^{5} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{256 c^{\frac{5}{2}}} + \frac{B c^{3} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*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)) + 2*A*a*c*Pi
ecewise((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)), (sq
rt(a)*x**6/6, True)) + A*c**2*Piecewise((-16*a**4*sqrt(a + c*x**2)/(315*c**4) +
8*a**3*x**2*sqrt(a + c*x**2)/(315*c**3) - 2*a**2*x**4*sqrt(a + c*x**2)/(105*c**2
) + a*x**6*sqrt(a + c*x**2)/(63*c) + x**8*sqrt(a + c*x**2)/9, Ne(c, 0)), (sqrt(a
)*x**8/8, True)) - 3*B*a**(9/2)*x/(256*c**2*sqrt(1 + c*x**2/a)) - B*a**(7/2)*x**
3/(256*c*sqrt(1 + c*x**2/a)) + 129*B*a**(5/2)*x**5/(640*sqrt(1 + c*x**2/a)) + 73
*B*a**(3/2)*c*x**7/(160*sqrt(1 + c*x**2/a)) + 29*B*sqrt(a)*c**2*x**9/(80*sqrt(1
+ c*x**2/a)) + 3*B*a**5*asinh(sqrt(c)*x/sqrt(a))/(256*c**(5/2)) + B*c**3*x**11/(
10*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.279523, size = 189, normalized size = 1.09 \[ -\frac{3 \, B a^{5}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{256 \, c^{\frac{5}{2}}} - \frac{1}{80640} \,{\left (\frac{2560 \, A a^{4}}{c^{2}} +{\left (\frac{945 \, B a^{4}}{c^{2}} - 2 \,{\left (\frac{640 \, A a^{3}}{c} +{\left (\frac{315 \, B a^{3}}{c} + 4 \,{\left (2400 \, A a^{2} +{\left (1953 \, B a^{2} + 2 \,{\left (1520 \, A a c + 7 \,{\left (189 \, B a c + 8 \,{\left (9 \, B c^{2} x + 10 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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