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

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

[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))/(1
60*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 = 0.104527, 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, Rules used = {833, 780, 195, 217, 206} \[ \frac{3 a^4 B x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}-\frac{a \left (a+c x^2\right )^{7/2} (160 A+189 B x)}{5040 c^2}+\frac{A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{B x^3 \left (a+c x^2\right )^{7/2}}{10 c} \]

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))/(1
60*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))

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^3 (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac{B x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{\int x^2 (-3 a B+10 A c x) \left (a+c x^2\right )^{5/2} \, dx}{10 c}\\ &=\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}+\frac{\int x (-20 a A c-27 a B c x) \left (a+c x^2\right )^{5/2} \, dx}{90 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}-\frac{a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (3 a^2 B\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{80 c^2}\\ &=\frac{a^2 B x \left (a+c x^2\right )^{5/2}}{160 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}-\frac{a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (a^3 B\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{32 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 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}-\frac{a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (3 a^4 B\right ) \int \sqrt{a+c x^2} \, dx}{128 c^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 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}-\frac{a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (3 a^5 B\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{256 c^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 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}-\frac{a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (3 a^5 B\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{256 c^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 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}-\frac{a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.272591, size = 145, normalized size = 0.84 \[ \frac{\sqrt{a+c x^2} \left (\sqrt{c} \left (24 a^2 c^2 x^4 (800 A+651 B x)+10 a^3 c x^2 (128 A+63 B x)-5 a^4 (512 A+189 B x)+16 a c^3 x^6 (1520 A+1323 B x)+896 c^4 x^8 (10 A+9 B x)\right )+\frac{945 a^{9/2} B \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{80640 c^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(Sqrt[c]*(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^(9/2)*B*ArcSinh[(Sqrt[c]*x)/
Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(80640*c^(5/2))

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Maple [A]  time = 0.007, size = 153, normalized size = 0.9 \begin{align*}{\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 ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\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}}}} \end{align*}

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]

1/10*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)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+1/9*A*x^
2*(c*x^2+a)^(7/2)/c-2/63*A*a/c^2*(c*x^2+a)^(7/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84923, size = 775, normalized size = 4.48 \begin{align*} \left [\frac{945 \, B a^{5} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (8064 \, B c^{5} x^{9} + 8960 \, A c^{5} x^{8} + 21168 \, B a c^{4} x^{7} + 24320 \, A a c^{4} x^{6} + 15624 \, B a^{2} c^{3} x^{5} + 19200 \, A a^{2} c^{3} x^{4} + 630 \, B a^{3} c^{2} x^{3} + 1280 \, A a^{3} c^{2} x^{2} - 945 \, B a^{4} c x - 2560 \, A a^{4} c\right )} \sqrt{c x^{2} + a}}{161280 \, c^{3}}, -\frac{945 \, B a^{5} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (8064 \, B c^{5} x^{9} + 8960 \, A c^{5} x^{8} + 21168 \, B a c^{4} x^{7} + 24320 \, A a c^{4} x^{6} + 15624 \, B a^{2} c^{3} x^{5} + 19200 \, A a^{2} c^{3} x^{4} + 630 \, B a^{3} c^{2} x^{3} + 1280 \, A a^{3} c^{2} x^{2} - 945 \, B a^{4} c x - 2560 \, A a^{4} c\right )} \sqrt{c x^{2} + a}}{80640 \, c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 43.638, size = 469, normalized size = 2.71 \begin{align*} 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}}} \end{align*}

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*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sq
rt(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)) + 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 [A]  time = 1.18725, size = 189, normalized size = 1.09 \begin{align*} -\frac{3 \, B a^{5} \log \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/256*B*a^5*log(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*(64
0*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)