3.80 \(\int x^2 (A+B x) (b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=203 \[ \frac{b^4 (b+2 c x) \sqrt{b x+c x^2} (9 b B-14 A c)}{1024 c^5}-\frac{b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-14 A c)}{384 c^4}-\frac{b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{b \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{120 c^3}-\frac{x \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c} \]

[Out]

(b^4*(9*b*B - 14*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) - (b^2*(9*b*B - 14*A*c)*(b + 2*c*x)*(b*x + c*x
^2)^(3/2))/(384*c^4) + (b*(9*b*B - 14*A*c)*(b*x + c*x^2)^(5/2))/(120*c^3) - ((9*b*B - 14*A*c)*x*(b*x + c*x^2)^
(5/2))/(84*c^2) + (B*x^2*(b*x + c*x^2)^(5/2))/(7*c) - (b^6*(9*b*B - 14*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x
^2]])/(1024*c^(11/2))

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Rubi [A]  time = 0.201263, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {794, 670, 640, 612, 620, 206} \[ \frac{b^4 (b+2 c x) \sqrt{b x+c x^2} (9 b B-14 A c)}{1024 c^5}-\frac{b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-14 A c)}{384 c^4}-\frac{b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{b \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{120 c^3}-\frac{x \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^4*(9*b*B - 14*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) - (b^2*(9*b*B - 14*A*c)*(b + 2*c*x)*(b*x + c*x
^2)^(3/2))/(384*c^4) + (b*(9*b*B - 14*A*c)*(b*x + c*x^2)^(5/2))/(120*c^3) - ((9*b*B - 14*A*c)*x*(b*x + c*x^2)^
(5/2))/(84*c^2) + (B*x^2*(b*x + c*x^2)^(5/2))/(7*c) - (b^6*(9*b*B - 14*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x
^2]])/(1024*c^(11/2))

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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^2 (A+B x) \left (b x+c x^2\right )^{3/2} \, dx &=\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (2 (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right ) \int x^2 \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=-\frac{(9 b B-14 A c) x \left (b x+c x^2\right )^{5/2}}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{(b (9 b B-14 A c)) \int x \left (b x+c x^2\right )^{3/2} \, dx}{24 c^2}\\ &=\frac{b (9 b B-14 A c) \left (b x+c x^2\right )^{5/2}}{120 c^3}-\frac{(9 b B-14 A c) x \left (b x+c x^2\right )^{5/2}}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}-\frac{\left (b^2 (9 b B-14 A c)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=-\frac{b^2 (9 b B-14 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{b (9 b B-14 A c) \left (b x+c x^2\right )^{5/2}}{120 c^3}-\frac{(9 b B-14 A c) x \left (b x+c x^2\right )^{5/2}}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (b^4 (9 b B-14 A c)\right ) \int \sqrt{b x+c x^2} \, dx}{256 c^4}\\ &=\frac{b^4 (9 b B-14 A c) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}-\frac{b^2 (9 b B-14 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{b (9 b B-14 A c) \left (b x+c x^2\right )^{5/2}}{120 c^3}-\frac{(9 b B-14 A c) x \left (b x+c x^2\right )^{5/2}}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}-\frac{\left (b^6 (9 b B-14 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2048 c^5}\\ &=\frac{b^4 (9 b B-14 A c) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}-\frac{b^2 (9 b B-14 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{b (9 b B-14 A c) \left (b x+c x^2\right )^{5/2}}{120 c^3}-\frac{(9 b B-14 A c) x \left (b x+c x^2\right )^{5/2}}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}-\frac{\left (b^6 (9 b B-14 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{1024 c^5}\\ &=\frac{b^4 (9 b B-14 A c) (b+2 c x) \sqrt{b x+c x^2}}{1024 c^5}-\frac{b^2 (9 b B-14 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac{b (9 b B-14 A c) \left (b x+c x^2\right )^{5/2}}{120 c^3}-\frac{(9 b B-14 A c) x \left (b x+c x^2\right )^{5/2}}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c}-\frac{b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.399771, size = 167, normalized size = 0.82 \[ \frac{x^4 \sqrt{x (b+c x)} \left (9 B (b+c x)^2-\frac{3 (9 b B-14 A c) \left (\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (-56 b^3 c^2 x^2+48 b^2 c^3 x^3+70 b^4 c x-105 b^5+1664 b c^4 x^4+1280 c^5 x^5\right )+105 b^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{5120 c^{9/2} x^{9/2} \sqrt{\frac{c x}{b}+1}}\right )}{63 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[x*(b + c*x)]*(9*B*(b + c*x)^2 - (3*(9*b*B - 14*A*c)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(-105*b^5 + 7
0*b^4*c*x - 56*b^3*c^2*x^2 + 48*b^2*c^3*x^3 + 1664*b*c^4*x^4 + 1280*c^5*x^5) + 105*b^(11/2)*ArcSinh[(Sqrt[c]*S
qrt[x])/Sqrt[b]]))/(5120*c^(9/2)*x^(9/2)*Sqrt[1 + (c*x)/b])))/(63*c)

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Maple [A]  time = 0.008, size = 327, normalized size = 1.6 \begin{align*}{\frac{B{x}^{2}}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{3\,bBx}{28\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{2}B}{40\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{3\,{b}^{3}Bx}{64\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{4}B}{128\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{9\,B{b}^{5}x}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{9\,B{b}^{6}}{1024\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{9\,B{b}^{7}}{2048}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}}+{\frac{Ax}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,Ab}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,A{b}^{2}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,A{b}^{3}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,A{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,A{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*x^2*(c*x^2+b*x)^(5/2)/c-3/28*B*b/c^2*x*(c*x^2+b*x)^(5/2)+3/40*B*b^2/c^3*(c*x^2+b*x)^(5/2)-3/64*B*b^3/c^3
*(c*x^2+b*x)^(3/2)*x-3/128*B*b^4/c^4*(c*x^2+b*x)^(3/2)+9/512*B*b^5/c^4*(c*x^2+b*x)^(1/2)*x+9/1024*B*b^6/c^5*(c
*x^2+b*x)^(1/2)-9/2048*B*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/6*A*x*(c*x^2+b*x)^(5/2)/c-7/
60*A*b/c^2*(c*x^2+b*x)^(5/2)+7/96*A*b^2/c^2*(c*x^2+b*x)^(3/2)*x+7/192*A*b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*A*b^4/
c^3*(c*x^2+b*x)^(1/2)*x-7/512*A*b^5/c^4*(c*x^2+b*x)^(1/2)+7/1024*A*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x)^(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^2*(B*x+A)*(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84598, size = 950, normalized size = 4.68 \begin{align*} \left [-\frac{105 \,{\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} + 1280 \,{\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \,{\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{4} - 48 \,{\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{3} + 56 \,{\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{2} - 70 \,{\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{215040 \, c^{6}}, \frac{105 \,{\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} + 1280 \,{\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 128 \,{\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{4} - 48 \,{\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{3} + 56 \,{\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{2} - 70 \,{\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{107520 \, c^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/215040*(105*(9*B*b^7 - 14*A*b^6*c)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(15360*B*c^7*x
^6 + 945*B*b^6*c - 1470*A*b^5*c^2 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 + 128*(3*B*b^2*c^5 + 182*A*b*c^6)*x^4 - 4
8*(9*B*b^3*c^4 - 14*A*b^2*c^5)*x^3 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^2 - 70*(9*B*b^5*c^2 - 14*A*b^4*c^3)*x)*
sqrt(c*x^2 + b*x))/c^6, 1/107520*(105*(9*B*b^7 - 14*A*b^6*c)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))
+ (15360*B*c^7*x^6 + 945*B*b^6*c - 1470*A*b^5*c^2 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 + 128*(3*B*b^2*c^5 + 182
*A*b*c^6)*x^4 - 48*(9*B*b^3*c^4 - 14*A*b^2*c^5)*x^3 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^2 - 70*(9*B*b^5*c^2 -
14*A*b^4*c^3)*x)*sqrt(c*x^2 + b*x))/c^6]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(x*(b + c*x))**(3/2)*(A + B*x), x)

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Giac [A]  time = 1.17846, size = 300, normalized size = 1.48 \begin{align*} \frac{1}{107520} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B c x + \frac{15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac{3 \, B b^{2} c^{5} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac{3 \,{\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )}}{c^{6}}\right )} x - \frac{35 \,{\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (9 \, B b^{6} c - 14 \, A b^{5} c^{2}\right )}}{c^{6}}\right )} + \frac{{\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(12*B*c*x + (15*B*b*c^6 + 14*A*c^7)/c^6)*x + (3*B*b^2*c^5 + 182*A*b
*c^6)/c^6)*x - 3*(9*B*b^3*c^4 - 14*A*b^2*c^5)/c^6)*x + 7*(9*B*b^4*c^3 - 14*A*b^3*c^4)/c^6)*x - 35*(9*B*b^5*c^2
- 14*A*b^4*c^3)/c^6)*x + 105*(9*B*b^6*c - 14*A*b^5*c^2)/c^6) + 1/2048*(9*B*b^7 - 14*A*b^6*c)*log(abs(-2*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)