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

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

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Rubi [A]  time = 0.20, antiderivative size = 203, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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])

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 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 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 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])

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*}

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Mathematica [A]  time = 0.34, size = 167, normalized size = 0.82 \begin {gather*} \frac {x^4 \sqrt {x (b+c x)} \left (9 B (b+c x)^2-\frac {3 (9 b B-14 A c) \left (105 b^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (-105 b^5+70 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\right )\right )}{5120 c^{9/2} x^{9/2} \sqrt {\frac {c x}{b}+1}}\right )}{63 c} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.75, size = 201, normalized size = 0.99 \begin {gather*} \frac {\left (9 b^7 B-14 A b^6 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{2048 c^{11/2}}+\frac {\sqrt {b x+c x^2} \left (-1470 A b^5 c+980 A b^4 c^2 x-784 A b^3 c^3 x^2+672 A b^2 c^4 x^3+23296 A b c^5 x^4+17920 A c^6 x^5+945 b^6 B-630 b^5 B c x+504 b^4 B c^2 x^2-432 b^3 B c^3 x^3+384 b^2 B c^4 x^4+19200 b B c^5 x^5+15360 B c^6 x^6\right )}{107520 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(945*b^6*B - 1470*A*b^5*c - 630*b^5*B*c*x + 980*A*b^4*c^2*x + 504*b^4*B*c^2*x^2 - 784*A*b^3
*c^3*x^2 - 432*b^3*B*c^3*x^3 + 672*A*b^2*c^4*x^3 + 384*b^2*B*c^4*x^4 + 23296*A*b*c^5*x^4 + 19200*b*B*c^5*x^5 +
17920*A*c^6*x^5 + 15360*B*c^6*x^6))/(107520*c^5) + ((9*b^7*B - 14*A*b^6*c)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x
+ c*x^2]])/(2048*c^(11/2))

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fricas [A]  time = 0.43, size = 399, normalized size = 1.97 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.24, size = 222, normalized size = 1.09 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.06, size = 327, normalized size = 1.61 \begin {gather*} \frac {7 A \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {9 B \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A \,b^{4} x}{256 c^{3}}+\frac {9 \sqrt {c \,x^{2}+b x}\, B \,b^{5} x}{512 c^{4}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A \,b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} x}{96 c^{2}}+\frac {9 \sqrt {c \,x^{2}+b x}\, B \,b^{6}}{1024 c^{5}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,x^{2}}{7 c}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A x}{6 c}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4}}{128 c^{4}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b x}{28 c^{2}}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b}{60 c^{2}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2}}{40 c^{3}} \end {gather*}

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((c*x+1/2*b)/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((c*x+1/2*b)/c^(1/2)+(c*x^2+b
*x)^(1/2))

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maxima [A]  time = 1.04, size = 324, normalized size = 1.60 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B x^{2}}{7 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{5} x}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} x}{64 \, c^{3}} - \frac {7 \, \sqrt {c x^{2} + b x} A b^{4} x}{256 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b x}{28 \, c^{2}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2} x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A x}{6 \, c} - \frac {9 \, B b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} + \frac {7 \, A b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{6}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4}}{128 \, c^{4}} - \frac {7 \, \sqrt {c x^{2} + b x} A b^{5}}{512 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2}}{40 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3}}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b}{60 \, c^{2}} \end {gather*}

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]

1/7*(c*x^2 + b*x)^(5/2)*B*x^2/c + 9/512*sqrt(c*x^2 + b*x)*B*b^5*x/c^4 - 3/64*(c*x^2 + b*x)^(3/2)*B*b^3*x/c^3 -
7/256*sqrt(c*x^2 + b*x)*A*b^4*x/c^3 - 3/28*(c*x^2 + b*x)^(5/2)*B*b*x/c^2 + 7/96*(c*x^2 + b*x)^(3/2)*A*b^2*x/c
^2 + 1/6*(c*x^2 + b*x)^(5/2)*A*x/c - 9/2048*B*b^7*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 7/10
24*A*b^6*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) + 9/1024*sqrt(c*x^2 + b*x)*B*b^6/c^5 - 3/128*(c*
x^2 + b*x)^(3/2)*B*b^4/c^4 - 7/512*sqrt(c*x^2 + b*x)*A*b^5/c^4 + 3/40*(c*x^2 + b*x)^(5/2)*B*b^2/c^3 + 7/192*(c
*x^2 + b*x)^(3/2)*A*b^3/c^3 - 7/60*(c*x^2 + b*x)^(5/2)*A*b/c^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )\, dx \end {gather*}

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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