3.1.94 \(\int x^2 (A+B x) (b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=241 \[ \frac {5 b^8 (11 b B-18 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{32768 c^{13/2}}-\frac {5 b^6 (b+2 c x) \sqrt {b x+c x^2} (11 b B-18 A c)}{32768 c^6}+\frac {5 b^4 (b+2 c x) \left (b x+c x^2\right )^{3/2} (11 b B-18 A c)}{12288 c^5}-\frac {b^2 (b+2 c x) \left (b x+c x^2\right )^{5/2} (11 b B-18 A c)}{768 c^4}+\frac {b \left (b x+c x^2\right )^{7/2} (11 b B-18 A c)}{224 c^3}-\frac {x \left (b x+c x^2\right )^{7/2} (11 b B-18 A c)}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c} \]

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Rubi [A]  time = 0.25, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {5 b^6 (b+2 c x) \sqrt {b x+c x^2} (11 b B-18 A c)}{32768 c^6}+\frac {5 b^4 (b+2 c x) \left (b x+c x^2\right )^{3/2} (11 b B-18 A c)}{12288 c^5}-\frac {b^2 (b+2 c x) \left (b x+c x^2\right )^{5/2} (11 b B-18 A c)}{768 c^4}+\frac {5 b^8 (11 b B-18 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{32768 c^{13/2}}+\frac {b \left (b x+c x^2\right )^{7/2} (11 b B-18 A c)}{224 c^3}-\frac {x \left (b x+c x^2\right )^{7/2} (11 b B-18 A c)}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b^6*(11*b*B - 18*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(32768*c^6) + (5*b^4*(11*b*B - 18*A*c)*(b + 2*c*x)*(b
*x + c*x^2)^(3/2))/(12288*c^5) - (b^2*(11*b*B - 18*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4) + (b*(11*b*
B - 18*A*c)*(b*x + c*x^2)^(7/2))/(224*c^3) - ((11*b*B - 18*A*c)*x*(b*x + c*x^2)^(7/2))/(144*c^2) + (B*x^2*(b*x
+ c*x^2)^(7/2))/(9*c) + (5*b^8*(11*b*B - 18*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(32768*c^(13/2))

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 )^{5/2} \, dx &=\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (2 (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right ) \int x^2 \left (b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {(b (11 b B-18 A c)) \int x \left (b x+c x^2\right )^{5/2} \, dx}{32 c^2}\\ &=\frac {b (11 b B-18 A c) \left (b x+c x^2\right )^{7/2}}{224 c^3}-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {\left (b^2 (11 b B-18 A c)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=-\frac {b^2 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {b (11 b B-18 A c) \left (b x+c x^2\right )^{7/2}}{224 c^3}-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (5 b^4 (11 b B-18 A c)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=\frac {5 b^4 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}-\frac {b^2 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {b (11 b B-18 A c) \left (b x+c x^2\right )^{7/2}}{224 c^3}-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}-\frac {\left (5 b^6 (11 b B-18 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{8192 c^5}\\ &=-\frac {5 b^6 (11 b B-18 A c) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}+\frac {5 b^4 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}-\frac {b^2 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {b (11 b B-18 A c) \left (b x+c x^2\right )^{7/2}}{224 c^3}-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (5 b^8 (11 b B-18 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{65536 c^6}\\ &=-\frac {5 b^6 (11 b B-18 A c) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}+\frac {5 b^4 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}-\frac {b^2 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {b (11 b B-18 A c) \left (b x+c x^2\right )^{7/2}}{224 c^3}-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (5 b^8 (11 b B-18 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{32768 c^6}\\ &=-\frac {5 b^6 (11 b B-18 A c) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}+\frac {5 b^4 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}-\frac {b^2 (11 b B-18 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {b (11 b B-18 A c) \left (b x+c x^2\right )^{7/2}}{224 c^3}-\frac {(11 b B-18 A c) x \left (b x+c x^2\right )^{7/2}}{144 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {5 b^8 (11 b B-18 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 197, normalized size = 0.82 \begin {gather*} \frac {x^3 (x (b+c x))^{5/2} \left (\frac {11 (11 b B-18 A c) \left (315 b^{15/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )-\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (315 b^7-210 b^6 c x+168 b^5 c^2 x^2-144 b^4 c^3 x^3+128 b^3 c^4 x^4+20736 b^2 c^5 x^5+33792 b c^6 x^6+14336 c^7 x^7\right )\right )}{229376 c^{11/2} x^{11/2} \sqrt {\frac {c x}{b}+1}}+11 B (b+c x)^3\right )}{99 c (b+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(x*(b + c*x))^(5/2)*(11*B*(b + c*x)^3 + (11*(11*b*B - 18*A*c)*(-(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(315*b
^7 - 210*b^6*c*x + 168*b^5*c^2*x^2 - 144*b^4*c^3*x^3 + 128*b^3*c^4*x^4 + 20736*b^2*c^5*x^5 + 33792*b*c^6*x^6 +
14336*c^7*x^7)) + 315*b^(15/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(229376*c^(11/2)*x^(11/2)*Sqrt[1 + (c*x)/
b])))/(99*c*(b + c*x)^2)

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IntegrateAlgebraic [A]  time = 0.93, size = 249, normalized size = 1.03 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (5670 A b^7 c-3780 A b^6 c^2 x+3024 A b^5 c^3 x^2-2592 A b^4 c^4 x^3+2304 A b^3 c^5 x^4+373248 A b^2 c^6 x^5+608256 A b c^7 x^6+258048 A c^8 x^7-3465 b^8 B+2310 b^7 B c x-1848 b^6 B c^2 x^2+1584 b^5 B c^3 x^3-1408 b^4 B c^4 x^4+1280 b^3 B c^5 x^5+316416 b^2 B c^6 x^6+530432 b B c^7 x^7+229376 B c^8 x^8\right )}{2064384 c^6}-\frac {5 \left (11 b^9 B-18 A b^8 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{65536 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[b*x + c*x^2]*(-3465*b^8*B + 5670*A*b^7*c + 2310*b^7*B*c*x - 3780*A*b^6*c^2*x - 1848*b^6*B*c^2*x^2 + 3024
*A*b^5*c^3*x^2 + 1584*b^5*B*c^3*x^3 - 2592*A*b^4*c^4*x^3 - 1408*b^4*B*c^4*x^4 + 2304*A*b^3*c^5*x^4 + 1280*b^3*
B*c^5*x^5 + 373248*A*b^2*c^6*x^5 + 316416*b^2*B*c^6*x^6 + 608256*A*b*c^7*x^6 + 530432*b*B*c^7*x^7 + 258048*A*c
^8*x^7 + 229376*B*c^8*x^8))/(2064384*c^6) - (5*(11*b^9*B - 18*A*b^8*c)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*
x^2]])/(65536*c^(13/2))

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fricas [A]  time = 0.43, size = 496, normalized size = 2.06 \begin {gather*} \left [-\frac {315 \, {\left (11 \, B b^{9} - 18 \, A b^{8} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (229376 \, B c^{9} x^{8} - 3465 \, B b^{8} c + 5670 \, A b^{7} c^{2} + 14336 \, {\left (37 \, B b c^{8} + 18 \, A c^{9}\right )} x^{7} + 3072 \, {\left (103 \, B b^{2} c^{7} + 198 \, A b c^{8}\right )} x^{6} + 256 \, {\left (5 \, B b^{3} c^{6} + 1458 \, A b^{2} c^{7}\right )} x^{5} - 128 \, {\left (11 \, B b^{4} c^{5} - 18 \, A b^{3} c^{6}\right )} x^{4} + 144 \, {\left (11 \, B b^{5} c^{4} - 18 \, A b^{4} c^{5}\right )} x^{3} - 168 \, {\left (11 \, B b^{6} c^{3} - 18 \, A b^{5} c^{4}\right )} x^{2} + 210 \, {\left (11 \, B b^{7} c^{2} - 18 \, A b^{6} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{4128768 \, c^{7}}, -\frac {315 \, {\left (11 \, B b^{9} - 18 \, A b^{8} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (229376 \, B c^{9} x^{8} - 3465 \, B b^{8} c + 5670 \, A b^{7} c^{2} + 14336 \, {\left (37 \, B b c^{8} + 18 \, A c^{9}\right )} x^{7} + 3072 \, {\left (103 \, B b^{2} c^{7} + 198 \, A b c^{8}\right )} x^{6} + 256 \, {\left (5 \, B b^{3} c^{6} + 1458 \, A b^{2} c^{7}\right )} x^{5} - 128 \, {\left (11 \, B b^{4} c^{5} - 18 \, A b^{3} c^{6}\right )} x^{4} + 144 \, {\left (11 \, B b^{5} c^{4} - 18 \, A b^{4} c^{5}\right )} x^{3} - 168 \, {\left (11 \, B b^{6} c^{3} - 18 \, A b^{5} c^{4}\right )} x^{2} + 210 \, {\left (11 \, B b^{7} c^{2} - 18 \, A b^{6} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{2064384 \, c^{7}}\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)^(5/2),x, algorithm="fricas")

[Out]

[-1/4128768*(315*(11*B*b^9 - 18*A*b^8*c)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(229376*B*c^
9*x^8 - 3465*B*b^8*c + 5670*A*b^7*c^2 + 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 + 3072*(103*B*b^2*c^7 + 198*A*b*c^8)
*x^6 + 256*(5*B*b^3*c^6 + 1458*A*b^2*c^7)*x^5 - 128*(11*B*b^4*c^5 - 18*A*b^3*c^6)*x^4 + 144*(11*B*b^5*c^4 - 18
*A*b^4*c^5)*x^3 - 168*(11*B*b^6*c^3 - 18*A*b^5*c^4)*x^2 + 210*(11*B*b^7*c^2 - 18*A*b^6*c^3)*x)*sqrt(c*x^2 + b*
x))/c^7, -1/2064384*(315*(11*B*b^9 - 18*A*b^8*c)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (229376*B
*c^9*x^8 - 3465*B*b^8*c + 5670*A*b^7*c^2 + 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 + 3072*(103*B*b^2*c^7 + 198*A*b*c
^8)*x^6 + 256*(5*B*b^3*c^6 + 1458*A*b^2*c^7)*x^5 - 128*(11*B*b^4*c^5 - 18*A*b^3*c^6)*x^4 + 144*(11*B*b^5*c^4 -
18*A*b^4*c^5)*x^3 - 168*(11*B*b^6*c^3 - 18*A*b^5*c^4)*x^2 + 210*(11*B*b^7*c^2 - 18*A*b^6*c^3)*x)*sqrt(c*x^2 +
b*x))/c^7]

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giac [A]  time = 0.46, size = 282, normalized size = 1.17 \begin {gather*} \frac {1}{2064384} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, B c^{2} x + \frac {37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac {3 \, {\left (103 \, B b^{2} c^{8} + 198 \, A b c^{9}\right )}}{c^{8}}\right )} x + \frac {5 \, B b^{3} c^{7} + 1458 \, A b^{2} c^{8}}{c^{8}}\right )} x - \frac {11 \, B b^{4} c^{6} - 18 \, A b^{3} c^{7}}{c^{8}}\right )} x + \frac {9 \, {\left (11 \, B b^{5} c^{5} - 18 \, A b^{4} c^{6}\right )}}{c^{8}}\right )} x - \frac {21 \, {\left (11 \, B b^{6} c^{4} - 18 \, A b^{5} c^{5}\right )}}{c^{8}}\right )} x + \frac {105 \, {\left (11 \, B b^{7} c^{3} - 18 \, A b^{6} c^{4}\right )}}{c^{8}}\right )} x - \frac {315 \, {\left (11 \, B b^{8} c^{2} - 18 \, A b^{7} c^{3}\right )}}{c^{8}}\right )} - \frac {5 \, {\left (11 \, B b^{9} - 18 \, A b^{8} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{65536 \, c^{\frac {13}{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)^(5/2),x, algorithm="giac")

[Out]

1/2064384*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*B*c^2*x + (37*B*b*c^9 + 18*A*c^10)/c^8)*x + 3*(103*B*b^2
*c^8 + 198*A*b*c^9)/c^8)*x + (5*B*b^3*c^7 + 1458*A*b^2*c^8)/c^8)*x - (11*B*b^4*c^6 - 18*A*b^3*c^7)/c^8)*x + 9*
(11*B*b^5*c^5 - 18*A*b^4*c^6)/c^8)*x - 21*(11*B*b^6*c^4 - 18*A*b^5*c^5)/c^8)*x + 105*(11*B*b^7*c^3 - 18*A*b^6*
c^4)/c^8)*x - 315*(11*B*b^8*c^2 - 18*A*b^7*c^3)/c^8) - 5/65536*(11*B*b^9 - 18*A*b^8*c)*log(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(13/2)

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maple [A]  time = 0.05, size = 409, normalized size = 1.70 \begin {gather*} -\frac {45 A \,b^{8} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}+\frac {55 B \,b^{9} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{65536 c^{\frac {13}{2}}}+\frac {45 \sqrt {c \,x^{2}+b x}\, A \,b^{6} x}{8192 c^{4}}-\frac {55 \sqrt {c \,x^{2}+b x}\, B \,b^{7} x}{16384 c^{5}}+\frac {45 \sqrt {c \,x^{2}+b x}\, A \,b^{7}}{16384 c^{5}}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{4} x}{1024 c^{3}}-\frac {55 \sqrt {c \,x^{2}+b x}\, B \,b^{8}}{32768 c^{6}}+\frac {55 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{5} x}{6144 c^{4}}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{5}}{2048 c^{4}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,b^{2} x}{64 c^{2}}+\frac {55 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{6}}{12288 c^{5}}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{3} x}{384 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} B \,x^{2}}{9 c}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,b^{3}}{128 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} A x}{8 c}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{4}}{768 c^{4}}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B b x}{144 c^{2}}-\frac {9 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} A b}{112 c^{2}}+\frac {11 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B \,b^{2}}{224 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)^(5/2),x)

[Out]

1/9*B*x^2*(c*x^2+b*x)^(7/2)/c-11/144*B*b/c^2*x*(c*x^2+b*x)^(7/2)+11/224*B*b^2/c^3*(c*x^2+b*x)^(7/2)-11/384*B*b
^3/c^3*(c*x^2+b*x)^(5/2)*x-11/768*B*b^4/c^4*(c*x^2+b*x)^(5/2)+55/6144*B*b^5/c^4*(c*x^2+b*x)^(3/2)*x+55/12288*B
*b^6/c^5*(c*x^2+b*x)^(3/2)-55/16384*B*b^7/c^5*(c*x^2+b*x)^(1/2)*x-55/32768*B*b^8/c^6*(c*x^2+b*x)^(1/2)+55/6553
6*B*b^9/c^(13/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/8*A*x*(c*x^2+b*x)^(7/2)/c-9/112*A*b/c^2*(c*x^2+b*
x)^(7/2)+3/64*A*b^2/c^2*(c*x^2+b*x)^(5/2)*x+3/128*A*b^3/c^3*(c*x^2+b*x)^(5/2)-15/1024*A*b^4/c^3*(c*x^2+b*x)^(3
/2)*x-15/2048*A*b^5/c^4*(c*x^2+b*x)^(3/2)+45/8192*A*b^6/c^4*(c*x^2+b*x)^(1/2)*x+45/16384*A*b^7/c^5*(c*x^2+b*x)
^(1/2)-45/32768*A*b^8/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))

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maxima [A]  time = 0.96, size = 406, normalized size = 1.68 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B x^{2}}{9 \, c} - \frac {55 \, \sqrt {c x^{2} + b x} B b^{7} x}{16384 \, c^{5}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5} x}{6144 \, c^{4}} + \frac {45 \, \sqrt {c x^{2} + b x} A b^{6} x}{8192 \, c^{4}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3} x}{384 \, c^{3}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{4} x}{1024 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b x}{144 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{2} x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} A x}{8 \, c} + \frac {55 \, B b^{9} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{65536 \, c^{\frac {13}{2}}} - \frac {45 \, A b^{8} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} - \frac {55 \, \sqrt {c x^{2} + b x} B b^{8}}{32768 \, c^{6}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{6}}{12288 \, c^{5}} + \frac {45 \, \sqrt {c x^{2} + b x} A b^{7}}{16384 \, c^{5}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{4}}{768 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{5}}{2048 \, c^{4}} + \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b^{2}}{224 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{3}}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} A b}{112 \, 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)^(5/2),x, algorithm="maxima")

[Out]

1/9*(c*x^2 + b*x)^(7/2)*B*x^2/c - 55/16384*sqrt(c*x^2 + b*x)*B*b^7*x/c^5 + 55/6144*(c*x^2 + b*x)^(3/2)*B*b^5*x
/c^4 + 45/8192*sqrt(c*x^2 + b*x)*A*b^6*x/c^4 - 11/384*(c*x^2 + b*x)^(5/2)*B*b^3*x/c^3 - 15/1024*(c*x^2 + b*x)^
(3/2)*A*b^4*x/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*B*b*x/c^2 + 3/64*(c*x^2 + b*x)^(5/2)*A*b^2*x/c^2 + 1/8*(c*x^2 +
b*x)^(7/2)*A*x/c + 55/65536*B*b^9*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(13/2) - 45/32768*A*b^8*log(
2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) - 55/32768*sqrt(c*x^2 + b*x)*B*b^8/c^6 + 55/12288*(c*x^2 + b
*x)^(3/2)*B*b^6/c^5 + 45/16384*sqrt(c*x^2 + b*x)*A*b^7/c^5 - 11/768*(c*x^2 + b*x)^(5/2)*B*b^4/c^4 - 15/2048*(c
*x^2 + b*x)^(3/2)*A*b^5/c^4 + 11/224*(c*x^2 + b*x)^(7/2)*B*b^2/c^3 + 3/128*(c*x^2 + b*x)^(5/2)*A*b^3/c^3 - 9/1
12*(c*x^2 + b*x)^(7/2)*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 )}^{5/2}\,\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)^(5/2)*(A + B*x),x)

[Out]

int(x^2*(b*x + c*x^2)^(5/2)*(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 {5}{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)**(5/2),x)

[Out]

Integral(x**2*(x*(b + c*x))**(5/2)*(A + B*x), x)

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