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

Optimal. Leaf size=276 \[ \frac{11 b^7 (b+2 c x) \sqrt{b x+c x^2} (13 b B-20 A c)}{131072 c^7}-\frac{11 b^5 (b+2 c x) \left (b x+c x^2\right )^{3/2} (13 b B-20 A c)}{49152 c^6}+\frac{11 b^3 (b+2 c x) \left (b x+c x^2\right )^{5/2} (13 b B-20 A c)}{15360 c^5}-\frac{11 b^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{4480 c^4}-\frac{11 b^9 (13 b B-20 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{131072 c^{15/2}}+\frac{11 b x \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{2880 c^3}-\frac{x^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c} \]

[Out]

(11*b^7*(13*b*B - 20*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(131072*c^7) - (11*b^5*(13*b*B - 20*A*c)*(b + 2*c*x)*
(b*x + c*x^2)^(3/2))/(49152*c^6) + (11*b^3*(13*b*B - 20*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(15360*c^5) - (1
1*b^2*(13*b*B - 20*A*c)*(b*x + c*x^2)^(7/2))/(4480*c^4) + (11*b*(13*b*B - 20*A*c)*x*(b*x + c*x^2)^(7/2))/(2880
*c^3) - ((13*b*B - 20*A*c)*x^2*(b*x + c*x^2)^(7/2))/(180*c^2) + (B*x^3*(b*x + c*x^2)^(7/2))/(10*c) - (11*b^9*(
13*b*B - 20*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(131072*c^(15/2))

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Rubi [A]  time = 0.298166, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{11 b^7 (b+2 c x) \sqrt{b x+c x^2} (13 b B-20 A c)}{131072 c^7}-\frac{11 b^5 (b+2 c x) \left (b x+c x^2\right )^{3/2} (13 b B-20 A c)}{49152 c^6}+\frac{11 b^3 (b+2 c x) \left (b x+c x^2\right )^{5/2} (13 b B-20 A c)}{15360 c^5}-\frac{11 b^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{4480 c^4}-\frac{11 b^9 (13 b B-20 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{131072 c^{15/2}}+\frac{11 b x \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{2880 c^3}-\frac{x^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*b^7*(13*b*B - 20*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(131072*c^7) - (11*b^5*(13*b*B - 20*A*c)*(b + 2*c*x)*
(b*x + c*x^2)^(3/2))/(49152*c^6) + (11*b^3*(13*b*B - 20*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(15360*c^5) - (1
1*b^2*(13*b*B - 20*A*c)*(b*x + c*x^2)^(7/2))/(4480*c^4) + (11*b*(13*b*B - 20*A*c)*x*(b*x + c*x^2)^(7/2))/(2880
*c^3) - ((13*b*B - 20*A*c)*x^2*(b*x + c*x^2)^(7/2))/(180*c^2) + (B*x^3*(b*x + c*x^2)^(7/2))/(10*c) - (11*b^9*(
13*b*B - 20*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(131072*c^(15/2))

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^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx &=\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}+\frac{\left (3 (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right ) \int x^3 \left (b x+c x^2\right )^{5/2} \, dx}{10 c}\\ &=-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}+\frac{(11 b (13 b B-20 A c)) \int x^2 \left (b x+c x^2\right )^{5/2} \, dx}{360 c^2}\\ &=\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac{\left (11 b^2 (13 b B-20 A c)\right ) \int x \left (b x+c x^2\right )^{5/2} \, dx}{640 c^3}\\ &=-\frac{11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}+\frac{\left (11 b^3 (13 b B-20 A c)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{1280 c^4}\\ &=\frac{11 b^3 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{15360 c^5}-\frac{11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac{\left (11 b^5 (13 b B-20 A c)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{6144 c^5}\\ &=-\frac{11 b^5 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{49152 c^6}+\frac{11 b^3 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{15360 c^5}-\frac{11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}+\frac{\left (11 b^7 (13 b B-20 A c)\right ) \int \sqrt{b x+c x^2} \, dx}{32768 c^6}\\ &=\frac{11 b^7 (13 b B-20 A c) (b+2 c x) \sqrt{b x+c x^2}}{131072 c^7}-\frac{11 b^5 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{49152 c^6}+\frac{11 b^3 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{15360 c^5}-\frac{11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac{\left (11 b^9 (13 b B-20 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{262144 c^7}\\ &=\frac{11 b^7 (13 b B-20 A c) (b+2 c x) \sqrt{b x+c x^2}}{131072 c^7}-\frac{11 b^5 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{49152 c^6}+\frac{11 b^3 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{15360 c^5}-\frac{11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac{\left (11 b^9 (13 b B-20 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{131072 c^7}\\ &=\frac{11 b^7 (13 b B-20 A c) (b+2 c x) \sqrt{b x+c x^2}}{131072 c^7}-\frac{11 b^5 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{49152 c^6}+\frac{11 b^3 (13 b B-20 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{15360 c^5}-\frac{11 b^2 (13 b B-20 A c) \left (b x+c x^2\right )^{7/2}}{4480 c^4}+\frac{11 b (13 b B-20 A c) x \left (b x+c x^2\right )^{7/2}}{2880 c^3}-\frac{(13 b B-20 A c) x^2 \left (b x+c x^2\right )^{7/2}}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c}-\frac{11 b^9 (13 b B-20 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{131072 c^{15/2}}\\ \end{align*}

Mathematica [A]  time = 0.372324, size = 207, normalized size = 0.75 \[ \frac{x^4 (x (b+c x))^{5/2} \left (13 B (b+c x)^3-\frac{13 (13 b B-20 A c) \left (\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (-1848 b^6 c^2 x^2+1584 b^5 c^3 x^3-1408 b^4 c^4 x^4+1280 b^3 c^5 x^5+316416 b^2 c^6 x^6+2310 b^7 c x-3465 b^8+530432 b c^7 x^7+229376 c^8 x^8\right )+3465 b^{17/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{4128768 c^{13/2} x^{13/2} \sqrt{\frac{c x}{b}+1}}\right )}{130 c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(x*(b + c*x))^(5/2)*(13*B*(b + c*x)^3 - (13*(13*b*B - 20*A*c)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(-3465*b
^8 + 2310*b^7*c*x - 1848*b^6*c^2*x^2 + 1584*b^5*c^3*x^3 - 1408*b^4*c^4*x^4 + 1280*b^3*c^5*x^5 + 316416*b^2*c^6
*x^6 + 530432*b*c^7*x^7 + 229376*c^8*x^8) + 3465*b^(17/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(4128768*c^(13/
2)*x^(13/2)*Sqrt[1 + (c*x)/b])))/(130*c*(b + c*x)^2)

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Maple [A]  time = 0.01, size = 455, normalized size = 1.7 \begin{align*}{\frac{B{x}^{3}}{10\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{13\,Bb{x}^{2}}{180\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{143\,{b}^{2}Bx}{2880\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{143\,{b}^{3}B}{4480\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{143\,{b}^{4}Bx}{7680\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{143\,B{b}^{5}}{15360\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{143\,{b}^{6}Bx}{24576\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{143\,B{b}^{7}}{49152\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{143\,B{b}^{8}x}{65536\,{c}^{6}}\sqrt{c{x}^{2}+bx}}+{\frac{143\,B{b}^{9}}{131072\,{c}^{7}}\sqrt{c{x}^{2}+bx}}-{\frac{143\,B{b}^{10}}{262144}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{15}{2}}}}+{\frac{A{x}^{2}}{9\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{11\,Abx}{144\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{11\,A{b}^{2}}{224\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{11\,A{b}^{3}x}{384\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{11\,A{b}^{4}}{768\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{55\,A{b}^{5}x}{6144\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{55\,A{b}^{6}}{12288\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{55\,A{b}^{7}x}{16384\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{55\,A{b}^{8}}{32768\,{c}^{6}}\sqrt{c{x}^{2}+bx}}+{\frac{55\,A{b}^{9}}{65536}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)*(c*x^2+b*x)^(5/2),x)

[Out]

1/10*B*x^3*(c*x^2+b*x)^(7/2)/c-13/180*B*b/c^2*x^2*(c*x^2+b*x)^(7/2)+143/2880*B*b^2/c^3*x*(c*x^2+b*x)^(7/2)-143
/4480*B*b^3/c^4*(c*x^2+b*x)^(7/2)+143/7680*B*b^4/c^4*(c*x^2+b*x)^(5/2)*x+143/15360*B*b^5/c^5*(c*x^2+b*x)^(5/2)
-143/24576*B*b^6/c^5*(c*x^2+b*x)^(3/2)*x-143/49152*B*b^7/c^6*(c*x^2+b*x)^(3/2)+143/65536*B*b^8/c^6*(c*x^2+b*x)
^(1/2)*x+143/131072*B*b^9/c^7*(c*x^2+b*x)^(1/2)-143/262144*B*b^10/c^(15/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^
(1/2))+1/9*A*x^2*(c*x^2+b*x)^(7/2)/c-11/144*A*b/c^2*x*(c*x^2+b*x)^(7/2)+11/224*A*b^2/c^3*(c*x^2+b*x)^(7/2)-11/
384*A*b^3/c^3*(c*x^2+b*x)^(5/2)*x-11/768*A*b^4/c^4*(c*x^2+b*x)^(5/2)+55/6144*A*b^5/c^4*(c*x^2+b*x)^(3/2)*x+55/
12288*A*b^6/c^5*(c*x^2+b*x)^(3/2)-55/16384*A*b^7/c^5*(c*x^2+b*x)^(1/2)*x-55/32768*A*b^8/c^6*(c*x^2+b*x)^(1/2)+
55/65536*A*b^9/c^(13/2)*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^3*(B*x+A)*(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.06951, size = 1353, normalized size = 4.9 \begin{align*} \left [-\frac{3465 \,{\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (4128768 \, B c^{10} x^{9} + 45045 \, B b^{9} c - 69300 \, A b^{8} c^{2} + 229376 \,{\left (41 \, B b c^{9} + 20 \, A c^{10}\right )} x^{8} + 14336 \,{\left (383 \, B b^{2} c^{8} + 740 \, A b c^{9}\right )} x^{7} + 15360 \,{\left (B b^{3} c^{7} + 412 \, A b^{2} c^{8}\right )} x^{6} - 1280 \,{\left (13 \, B b^{4} c^{6} - 20 \, A b^{3} c^{7}\right )} x^{5} + 1408 \,{\left (13 \, B b^{5} c^{5} - 20 \, A b^{4} c^{6}\right )} x^{4} - 1584 \,{\left (13 \, B b^{6} c^{4} - 20 \, A b^{5} c^{5}\right )} x^{3} + 1848 \,{\left (13 \, B b^{7} c^{3} - 20 \, A b^{6} c^{4}\right )} x^{2} - 2310 \,{\left (13 \, B b^{8} c^{2} - 20 \, A b^{7} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{82575360 \, c^{8}}, \frac{3465 \,{\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (4128768 \, B c^{10} x^{9} + 45045 \, B b^{9} c - 69300 \, A b^{8} c^{2} + 229376 \,{\left (41 \, B b c^{9} + 20 \, A c^{10}\right )} x^{8} + 14336 \,{\left (383 \, B b^{2} c^{8} + 740 \, A b c^{9}\right )} x^{7} + 15360 \,{\left (B b^{3} c^{7} + 412 \, A b^{2} c^{8}\right )} x^{6} - 1280 \,{\left (13 \, B b^{4} c^{6} - 20 \, A b^{3} c^{7}\right )} x^{5} + 1408 \,{\left (13 \, B b^{5} c^{5} - 20 \, A b^{4} c^{6}\right )} x^{4} - 1584 \,{\left (13 \, B b^{6} c^{4} - 20 \, A b^{5} c^{5}\right )} x^{3} + 1848 \,{\left (13 \, B b^{7} c^{3} - 20 \, A b^{6} c^{4}\right )} x^{2} - 2310 \,{\left (13 \, B b^{8} c^{2} - 20 \, A b^{7} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{41287680 \, c^{8}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/82575360*(3465*(13*B*b^10 - 20*A*b^9*c)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(4128768*
B*c^10*x^9 + 45045*B*b^9*c - 69300*A*b^8*c^2 + 229376*(41*B*b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 74
0*A*b*c^9)*x^7 + 15360*(B*b^3*c^7 + 412*A*b^2*c^8)*x^6 - 1280*(13*B*b^4*c^6 - 20*A*b^3*c^7)*x^5 + 1408*(13*B*b
^5*c^5 - 20*A*b^4*c^6)*x^4 - 1584*(13*B*b^6*c^4 - 20*A*b^5*c^5)*x^3 + 1848*(13*B*b^7*c^3 - 20*A*b^6*c^4)*x^2 -
2310*(13*B*b^8*c^2 - 20*A*b^7*c^3)*x)*sqrt(c*x^2 + b*x))/c^8, 1/41287680*(3465*(13*B*b^10 - 20*A*b^9*c)*sqrt(
-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (4128768*B*c^10*x^9 + 45045*B*b^9*c - 69300*A*b^8*c^2 + 229376*
(41*B*b*c^9 + 20*A*c^10)*x^8 + 14336*(383*B*b^2*c^8 + 740*A*b*c^9)*x^7 + 15360*(B*b^3*c^7 + 412*A*b^2*c^8)*x^6
- 1280*(13*B*b^4*c^6 - 20*A*b^3*c^7)*x^5 + 1408*(13*B*b^5*c^5 - 20*A*b^4*c^6)*x^4 - 1584*(13*B*b^6*c^4 - 20*A
*b^5*c^5)*x^3 + 1848*(13*B*b^7*c^3 - 20*A*b^6*c^4)*x^2 - 2310*(13*B*b^8*c^2 - 20*A*b^7*c^3)*x)*sqrt(c*x^2 + b*
x))/c^8]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

Integral(x**3*(x*(b + c*x))**(5/2)*(A + B*x), x)

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Giac [A]  time = 1.17972, size = 417, normalized size = 1.51 \begin{align*} \frac{1}{41287680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \,{\left (18 \, B c^{2} x + \frac{41 \, B b c^{10} + 20 \, A c^{11}}{c^{9}}\right )} x + \frac{383 \, B b^{2} c^{9} + 740 \, A b c^{10}}{c^{9}}\right )} x + \frac{15 \,{\left (B b^{3} c^{8} + 412 \, A b^{2} c^{9}\right )}}{c^{9}}\right )} x - \frac{5 \,{\left (13 \, B b^{4} c^{7} - 20 \, A b^{3} c^{8}\right )}}{c^{9}}\right )} x + \frac{11 \,{\left (13 \, B b^{5} c^{6} - 20 \, A b^{4} c^{7}\right )}}{c^{9}}\right )} x - \frac{99 \,{\left (13 \, B b^{6} c^{5} - 20 \, A b^{5} c^{6}\right )}}{c^{9}}\right )} x + \frac{231 \,{\left (13 \, B b^{7} c^{4} - 20 \, A b^{6} c^{5}\right )}}{c^{9}}\right )} x - \frac{1155 \,{\left (13 \, B b^{8} c^{3} - 20 \, A b^{7} c^{4}\right )}}{c^{9}}\right )} x + \frac{3465 \,{\left (13 \, B b^{9} c^{2} - 20 \, A b^{8} c^{3}\right )}}{c^{9}}\right )} + \frac{11 \,{\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{262144 \, c^{\frac{15}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/41287680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*(18*B*c^2*x + (41*B*b*c^10 + 20*A*c^11)/c^9)*x + (383*B
*b^2*c^9 + 740*A*b*c^10)/c^9)*x + 15*(B*b^3*c^8 + 412*A*b^2*c^9)/c^9)*x - 5*(13*B*b^4*c^7 - 20*A*b^3*c^8)/c^9)
*x + 11*(13*B*b^5*c^6 - 20*A*b^4*c^7)/c^9)*x - 99*(13*B*b^6*c^5 - 20*A*b^5*c^6)/c^9)*x + 231*(13*B*b^7*c^4 - 2
0*A*b^6*c^5)/c^9)*x - 1155*(13*B*b^8*c^3 - 20*A*b^7*c^4)/c^9)*x + 3465*(13*B*b^9*c^2 - 20*A*b^8*c^3)/c^9) + 11
/262144*(13*B*b^10 - 20*A*b^9*c)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(15/2)