3.326 \(\int x^4 (A+B x) (a+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=175 \[ \frac{3 a^3 A x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{3 a^4 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}+\frac{a \left (a+c x^2\right )^{5/2} (128 a B-315 A c x)}{5040 c^3}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c} \]

[Out]

(3*a^3*A*x*Sqrt[a + c*x^2])/(128*c^2) + (a^2*A*x*(a + c*x^2)^(3/2))/(64*c^2) - (4*a*B*x^2*(a + c*x^2)^(5/2))/(
63*c^2) + (A*x^3*(a + c*x^2)^(5/2))/(8*c) + (B*x^4*(a + c*x^2)^(5/2))/(9*c) + (a*(128*a*B - 315*A*c*x)*(a + c*
x^2)^(5/2))/(5040*c^3) + (3*a^4*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(5/2))

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Rubi [A]  time = 0.135107, antiderivative size = 175, 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} \[ \frac{3 a^3 A x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}+\frac{3 a^4 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}+\frac{a \left (a+c x^2\right )^{5/2} (128 a B-315 A c x)}{5040 c^3}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*(a + c*x^2)^(3/2),x]

[Out]

(3*a^3*A*x*Sqrt[a + c*x^2])/(128*c^2) + (a^2*A*x*(a + c*x^2)^(3/2))/(64*c^2) - (4*a*B*x^2*(a + c*x^2)^(5/2))/(
63*c^2) + (A*x^3*(a + c*x^2)^(5/2))/(8*c) + (B*x^4*(a + c*x^2)^(5/2))/(9*c) + (a*(128*a*B - 315*A*c*x)*(a + c*
x^2)^(5/2))/(5040*c^3) + (3*a^4*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*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^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{\int x^3 (-4 a B+9 A c x) \left (a+c x^2\right )^{3/2} \, dx}{9 c}\\ &=\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{\int x^2 (-27 a A c-32 a B c x) \left (a+c x^2\right )^{3/2} \, dx}{72 c^2}\\ &=-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{\int x \left (64 a^2 B c-189 a A c^2 x\right ) \left (a+c x^2\right )^{3/2} \, dx}{504 c^3}\\ &=-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac{\left (a^2 A\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{16 c^2}\\ &=\frac{a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac{\left (3 a^3 A\right ) \int \sqrt{a+c x^2} \, dx}{64 c^2}\\ &=\frac{3 a^3 A x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac{\left (3 a^4 A\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c^2}\\ &=\frac{3 a^3 A x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac{\left (3 a^4 A\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c^2}\\ &=\frac{3 a^3 A x \sqrt{a+c x^2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac{A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac{B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac{a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac{3 a^4 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.25439, size = 132, normalized size = 0.75 \[ \frac{\sqrt{a+c x^2} \left (6 a^2 c^2 x^3 (105 A+64 B x)-a^3 c x (945 A+512 B x)+\frac{945 a^{7/2} A \sqrt{c} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}+1024 a^4 B+40 a c^3 x^5 (189 A+160 B x)+560 c^4 x^7 (9 A+8 B x)\right )}{40320 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*(a + c*x^2)^(3/2),x]

[Out]

(Sqrt[a + c*x^2]*(1024*a^4*B + 560*c^4*x^7*(9*A + 8*B*x) + 6*a^2*c^2*x^3*(105*A + 64*B*x) + 40*a*c^3*x^5*(189*
A + 160*B*x) - a^3*c*x*(945*A + 512*B*x) + (945*a^(7/2)*A*Sqrt[c]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^
2)/a]))/(40320*c^3)

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Maple [A]  time = 0.01, size = 155, normalized size = 0.9 \begin{align*}{\frac{B{x}^{4}}{9\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{4\,aB{x}^{2}}{63\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{8\,B{a}^{2}}{315\,{c}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{x}^{3}}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aAx}{16\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Ax}{64\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}Ax}{128\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,A{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(c*x^2+a)^(3/2),x)

[Out]

1/9*B*x^4*(c*x^2+a)^(5/2)/c-4/63*a*B*x^2*(c*x^2+a)^(5/2)/c^2+8/315*B*a^2/c^3*(c*x^2+a)^(5/2)+1/8*A*x^3*(c*x^2+
a)^(5/2)/c-1/16*A*a/c^2*x*(c*x^2+a)^(5/2)+1/64*a^2*A*x*(c*x^2+a)^(3/2)/c^2+3/128*a^3*A*x*(c*x^2+a)^(1/2)/c^2+3
/128*A*a^4/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

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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^4*(B*x+A)*(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76278, size = 690, normalized size = 3.94 \begin{align*} \left [\frac{945 \, A a^{4} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (4480 \, B c^{4} x^{8} + 5040 \, A c^{4} x^{7} + 6400 \, B a c^{3} x^{6} + 7560 \, A a c^{3} x^{5} + 384 \, B a^{2} c^{2} x^{4} + 630 \, A a^{2} c^{2} x^{3} - 512 \, B a^{3} c x^{2} - 945 \, A a^{3} c x + 1024 \, B a^{4}\right )} \sqrt{c x^{2} + a}}{80640 \, c^{3}}, -\frac{945 \, A a^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (4480 \, B c^{4} x^{8} + 5040 \, A c^{4} x^{7} + 6400 \, B a c^{3} x^{6} + 7560 \, A a c^{3} x^{5} + 384 \, B a^{2} c^{2} x^{4} + 630 \, A a^{2} c^{2} x^{3} - 512 \, B a^{3} c x^{2} - 945 \, A a^{3} c x + 1024 \, B a^{4}\right )} \sqrt{c x^{2} + a}}{40320 \, c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[1/80640*(945*A*a^4*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(4480*B*c^4*x^8 + 5040*A*c^4*x
^7 + 6400*B*a*c^3*x^6 + 7560*A*a*c^3*x^5 + 384*B*a^2*c^2*x^4 + 630*A*a^2*c^2*x^3 - 512*B*a^3*c*x^2 - 945*A*a^3
*c*x + 1024*B*a^4)*sqrt(c*x^2 + a))/c^3, -1/40320*(945*A*a^4*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (44
80*B*c^4*x^8 + 5040*A*c^4*x^7 + 6400*B*a*c^3*x^6 + 7560*A*a*c^3*x^5 + 384*B*a^2*c^2*x^4 + 630*A*a^2*c^2*x^3 -
512*B*a^3*c*x^2 - 945*A*a^3*c*x + 1024*B*a^4)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]  time = 20.5333, size = 366, normalized size = 2.09 \begin{align*} - \frac{3 A a^{\frac{7}{2}} x}{128 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{\frac{5}{2}} x^{3}}{128 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{13 A a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 A \sqrt{a} c x^{7}}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A a^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{128 c^{\frac{5}{2}}} + \frac{A c^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a \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 ) + B c \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(c*x**2+a)**(3/2),x)

[Out]

-3*A*a**(7/2)*x/(128*c**2*sqrt(1 + c*x**2/a)) - A*a**(5/2)*x**3/(128*c*sqrt(1 + c*x**2/a)) + 13*A*a**(3/2)*x**
5/(64*sqrt(1 + c*x**2/a)) + 5*A*sqrt(a)*c*x**7/(16*sqrt(1 + c*x**2/a)) + 3*A*a**4*asinh(sqrt(c)*x/sqrt(a))/(12
8*c**(5/2)) + A*c**2*x**9/(8*sqrt(a)*sqrt(1 + c*x**2/a)) + B*a*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)) + B*c*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))

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Giac [A]  time = 1.19011, size = 176, normalized size = 1.01 \begin{align*} -\frac{3 \, A a^{4} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{5}{2}}} + \frac{1}{40320} \, \sqrt{c x^{2} + a}{\left (\frac{1024 \, B a^{4}}{c^{3}} -{\left (\frac{945 \, A a^{3}}{c^{2}} + 2 \,{\left (\frac{256 \, B a^{3}}{c^{2}} -{\left (\frac{315 \, A a^{2}}{c} + 4 \,{\left (\frac{48 \, B a^{2}}{c} + 5 \,{\left (189 \, A a + 2 \,{\left (80 \, B a + 7 \,{\left (8 \, B c x + 9 \, A c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

-3/128*A*a^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/40320*sqrt(c*x^2 + a)*(1024*B*a^4/c^3 - (945*A
*a^3/c^2 + 2*(256*B*a^3/c^2 - (315*A*a^2/c + 4*(48*B*a^2/c + 5*(189*A*a + 2*(80*B*a + 7*(8*B*c*x + 9*A*c)*x)*x
)*x)*x)*x)*x)*x)