3.1.93 \(\int x^3 (A+B x) (b x+c x^2)^{5/2} \, dx\) [93]

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

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 276, normalized size of antiderivative = 1.00, 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 = {808, 684, 654, 626, 634, 212} \begin {gather*} -\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^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 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} \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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 634

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 654

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 684

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 808

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^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 ) \text {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*}

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Mathematica [A]
time = 0.60, size = 258, normalized size = 0.93 \begin {gather*} \frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (45045 b^9 B+5120 b^3 c^6 x^5 (5 A+3 B x)+458752 c^9 x^8 (10 A+9 B x)-1280 b^4 c^5 x^4 (22 A+13 B x)+1848 b^7 c^2 x (25 A+13 B x)-2310 b^8 c (30 A+13 B x)+704 b^5 c^4 x^3 (45 A+26 B x)-528 b^6 c^3 x^2 (70 A+39 B x)+57344 b c^8 x^7 (185 A+164 B x)+2048 b^2 c^7 x^6 (3090 A+2681 B x)\right )+3465 b^9 (13 b B-20 A c) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{41287680 c^{15/2} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(45045*b^9*B + 5120*b^3*c^6*x^5*(5*A + 3*B*x) + 458752*c
^9*x^8*(10*A + 9*B*x) - 1280*b^4*c^5*x^4*(22*A + 13*B*x) + 1848*b^7*c^2*x*(25*A + 13*B*x) - 2310*b^8*c*(30*A +
13*B*x) + 704*b^5*c^4*x^3*(45*A + 26*B*x) - 528*b^6*c^3*x^2*(70*A + 39*B*x) + 57344*b*c^8*x^7*(185*A + 164*B*
x) + 2048*b^2*c^7*x^6*(3090*A + 2681*B*x)) + 3465*b^9*(13*b*B - 20*A*c)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]
]))/(41287680*c^(15/2)*Sqrt[x*(b + c*x)])

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Maple [A]
time = 0.54, size = 412, normalized size = 1.49

method	result	size
risch	\(-\frac {\left (-4128768 B \,c^{9} x^{9}-4587520 A \,c^{9} x^{8}-9404416 B b \,c^{8} x^{8}-10608640 A b \,c^{8} x^{7}-5490688 B \,b^{2} c^{7} x^{7}-6328320 A \,b^{2} c^{7} x^{6}-15360 B \,b^{3} c^{6} x^{6}-25600 A \,b^{3} c^{6} x^{5}+16640 B \,b^{4} c^{5} x^{5}+28160 A \,b^{4} c^{5} x^{4}-18304 B \,b^{5} c^{4} x^{4}-31680 A \,b^{5} c^{4} x^{3}+20592 B \,b^{6} c^{3} x^{3}+36960 A \,b^{6} c^{3} x^{2}-24024 B \,b^{7} c^{2} x^{2}-46200 A \,b^{7} c^{2} x +30030 B \,b^{8} c x +69300 A \,b^{8} c -45045 B \,b^{9}\right ) x \left (c x +b \right )}{41287680 c^{7} \sqrt {x \left (c x +b \right )}}+\frac {55 b^{9} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) A}{65536 c^{\frac {13}{2}}}-\frac {143 b^{10} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) B}{262144 c^{\frac {15}{2}}}\)	\(290\)
default	\(B \left (\frac {x^{3} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{10 c}-\frac {13 b \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )}{20 c}\right )+A \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 c}-\frac {11 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )}{18 c}\right )\)	\(412\)

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,method=_RETURNVERBOSE)

[Out]

B*(1/10*x^3*(c*x^2+b*x)^(7/2)/c-13/20*b/c*(1/9*x^2*(c*x^2+b*x)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x)^(7/2)/c-9/
16*b/c*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b
*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(
1/2)))))))))+A*(1/9*x^2*(c*x^2+b*x)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x)^(7/
2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4
*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))))))))

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Maxima [A]
time = 0.28, size = 452, normalized size = 1.64 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B x^{3}}{10 \, c} - \frac {13 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b x^{2}}{180 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} A x^{2}}{9 \, c} + \frac {143 \, \sqrt {c x^{2} + b x} B b^{8} x}{65536 \, c^{6}} - \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{6} x}{24576 \, c^{5}} - \frac {55 \, \sqrt {c x^{2} + b x} A b^{7} x}{16384 \, c^{5}} + \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{4} x}{7680 \, c^{4}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{5} x}{6144 \, c^{4}} + \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b^{2} x}{2880 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{3} x}{384 \, c^{3}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} A b x}{144 \, c^{2}} - \frac {143 \, B b^{10} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{262144 \, c^{\frac {15}{2}}} + \frac {55 \, A b^{9} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{65536 \, c^{\frac {13}{2}}} + \frac {143 \, \sqrt {c x^{2} + b x} B b^{9}}{131072 \, c^{7}} - \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{7}}{49152 \, c^{6}} - \frac {55 \, \sqrt {c x^{2} + b x} A b^{8}}{32768 \, c^{6}} + \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{5}}{15360 \, c^{5}} + \frac {55 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{6}}{12288 \, c^{5}} - \frac {143 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b^{3}}{4480 \, c^{4}} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{4}}{768 \, c^{4}} + \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} A b^{2}}{224 \, c^{3}} \end {gather*}

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]

1/10*(c*x^2 + b*x)^(7/2)*B*x^3/c - 13/180*(c*x^2 + b*x)^(7/2)*B*b*x^2/c^2 + 1/9*(c*x^2 + b*x)^(7/2)*A*x^2/c +
143/65536*sqrt(c*x^2 + b*x)*B*b^8*x/c^6 - 143/24576*(c*x^2 + b*x)^(3/2)*B*b^6*x/c^5 - 55/16384*sqrt(c*x^2 + b*
x)*A*b^7*x/c^5 + 143/7680*(c*x^2 + b*x)^(5/2)*B*b^4*x/c^4 + 55/6144*(c*x^2 + b*x)^(3/2)*A*b^5*x/c^4 + 143/2880
*(c*x^2 + b*x)^(7/2)*B*b^2*x/c^3 - 11/384*(c*x^2 + b*x)^(5/2)*A*b^3*x/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*A*b*x/c
^2 - 143/262144*B*b^10*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(15/2) + 55/65536*A*b^9*log(2*c*x + b +
2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(13/2) + 143/131072*sqrt(c*x^2 + b*x)*B*b^9/c^7 - 143/49152*(c*x^2 + b*x)^(3/2)
*B*b^7/c^6 - 55/32768*sqrt(c*x^2 + b*x)*A*b^8/c^6 + 143/15360*(c*x^2 + b*x)^(5/2)*B*b^5/c^5 + 55/12288*(c*x^2
+ b*x)^(3/2)*A*b^6/c^5 - 143/4480*(c*x^2 + b*x)^(7/2)*B*b^3/c^4 - 11/768*(c*x^2 + b*x)^(5/2)*A*b^4/c^4 + 11/22
4*(c*x^2 + b*x)^(7/2)*A*b^2/c^3

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Fricas [A]
time = 2.86, size = 541, normalized size = 1.96 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \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**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 = 0.78, size = 309, normalized size = 1.12 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\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^3*(b*x + c*x^2)^(5/2)*(A + B*x),x)

[Out]

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

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