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

Optimal. Leaf size=423 \[ \frac {\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac {5 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}+\frac {B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

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Rubi [A]  time = 0.43, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {832, 779, 612, 620, 206} \begin {gather*} \frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac {\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac {5 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac {B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^4*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*Sqrt[b*
x + c*x^2])/(32768*c^6) - (5*b^2*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d + 2
*A*e))*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(12288*c^5) + ((64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e)
- 32*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4) + (B*(d + e*x)^2*(b*x + c*x^2)^(7/2))/
(9*c) + ((18*A*c*e*(32*c*d - 9*b*e) + B*(64*c^2*d^2 - 324*b*c*d*e + 99*b^2*e^2) + 14*c*e*(4*B*c*d - 11*b*B*e +
18*A*c*e)*x)*(b*x + c*x^2)^(7/2))/(2016*c^3) - (5*b^6*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e)
- 32*b*c^2*d*(B*d + 2*A*e))*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 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

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

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\int (d+e x) \left (-\frac {1}{2} (7 b B-18 A c) d+\frac {1}{2} (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{8192 c^5}\\ &=\frac {5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{65536 c^6}\\ &=\frac {5 b^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right )\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^4 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{32768 c^6}-\frac {5 b^2 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {\left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{768 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (18 A c e (32 c d-9 b e)+B \left (64 c^2 d^2-324 b c d e+99 b^2 e^2\right )+14 c e (4 B c d-11 b B e+18 A c e) x\right ) \left (b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 b^6 \left (64 A c^3 d^2-11 b^3 B e^2+18 b^2 c e (2 B d+A e)-32 b c^2 d (B d+2 A e)\right ) \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 = 1.45, size = 417, normalized size = 0.99 \begin {gather*} \frac {(x (b+c x))^{7/2} \left (\frac {1323 A \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \left (\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right )-15 b^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )}{1024 c^{9/2} x^{7/2} \sqrt {\frac {c x}{b}+1}}+\frac {5103 A e (b+c x)^3 (2 c d-b e)}{c}+7938 A e (b+c x)^3 (d+e x)+\frac {189 B \left (11 b^2 e^2-36 b c d e+32 c^2 d^2\right ) \left (105 b^{13/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (-105 b^6+70 b^5 c x-56 b^4 c^2 x^2+48 b^3 c^3 x^3+4736 b^2 c^4 x^4+7424 b c^5 x^5+3072 c^6 x^6\right )\right )}{2048 c^{11/2} x^{7/2} \sqrt {\frac {c x}{b}+1}}+\frac {441 B e x (b+c x)^3 (20 c d-11 b e)}{c}+7056 B e x (b+c x)^3 (d+e x)\right )}{63504 c (b+c x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(7/2)*((5103*A*e*(2*c*d - b*e)*(b + c*x)^3)/c + (441*B*e*(20*c*d - 11*b*e)*x*(b + c*x)^3)/c + 7
938*A*e*(b + c*x)^3*(d + e*x) + 7056*B*e*x*(b + c*x)^3*(d + e*x) + (1323*A*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^
2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^2*c^3*x^3 + 640*b*c^4*x^4 +
256*c^5*x^5) - 15*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(1024*c^(9/2)*x^(7/2)*Sqrt[1 + (c*x)/b]) + (1
89*B*(32*c^2*d^2 - 36*b*c*d*e + 11*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(-105*b^6 + 70*b^5*c*x - 56*b^4
*c^2*x^2 + 48*b^3*c^3*x^3 + 4736*b^2*c^4*x^4 + 7424*b*c^5*x^5 + 3072*c^6*x^6) + 105*b^(13/2)*ArcSinh[(Sqrt[c]*
Sqrt[x])/Sqrt[b]]))/(2048*c^(11/2)*x^(7/2)*Sqrt[1 + (c*x)/b])))/(63504*c*(b + c*x)^3)

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IntegrateAlgebraic [A]  time = 3.28, size = 713, normalized size = 1.69 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (5670 A b^7 c e^2-20160 A b^6 c^2 d e-3780 A b^6 c^2 e^2 x+20160 A b^5 c^3 d^2+13440 A b^5 c^3 d e x+3024 A b^5 c^3 e^2 x^2-13440 A b^4 c^4 d^2 x-10752 A b^4 c^4 d e x^2-2592 A b^4 c^4 e^2 x^3+10752 A b^3 c^5 d^2 x^2+9216 A b^3 c^5 d e x^3+2304 A b^3 c^5 e^2 x^4+580608 A b^2 c^6 d^2 x^3+909312 A b^2 c^6 d e x^4+373248 A b^2 c^6 e^2 x^5+860160 A b c^7 d^2 x^4+1425408 A b c^7 d e x^5+608256 A b c^7 e^2 x^6+344064 A c^8 d^2 x^5+589824 A c^8 d e x^6+258048 A c^8 e^2 x^7-3465 b^8 B e^2+11340 b^7 B c d e+2310 b^7 B c e^2 x-10080 b^6 B c^2 d^2-7560 b^6 B c^2 d e x-1848 b^6 B c^2 e^2 x^2+6720 b^5 B c^3 d^2 x+6048 b^5 B c^3 d e x^2+1584 b^5 B c^3 e^2 x^3-5376 b^4 B c^4 d^2 x^2-5184 b^4 B c^4 d e x^3-1408 b^4 B c^4 e^2 x^4+4608 b^3 B c^5 d^2 x^3+4608 b^3 B c^5 d e x^4+1280 b^3 B c^5 e^2 x^5+454656 b^2 B c^6 d^2 x^4+746496 b^2 B c^6 d e x^5+316416 b^2 B c^6 e^2 x^6+712704 b B c^7 d^2 x^5+1216512 b B c^7 d e x^6+530432 b B c^7 e^2 x^7+294912 B c^8 d^2 x^6+516096 B c^8 d e x^7+229376 B c^8 e^2 x^8\right )}{2064384 c^6}-\frac {5 \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-18 A b^8 c e^2+64 A b^7 c^2 d e-64 A b^6 c^3 d^2+11 b^9 B e^2-36 b^8 B c d e+32 b^7 B c^2 d^2\right )}{65536 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(-10080*b^6*B*c^2*d^2 + 20160*A*b^5*c^3*d^2 + 11340*b^7*B*c*d*e - 20160*A*b^6*c^2*d*e - 346
5*b^8*B*e^2 + 5670*A*b^7*c*e^2 + 6720*b^5*B*c^3*d^2*x - 13440*A*b^4*c^4*d^2*x - 7560*b^6*B*c^2*d*e*x + 13440*A
*b^5*c^3*d*e*x + 2310*b^7*B*c*e^2*x - 3780*A*b^6*c^2*e^2*x - 5376*b^4*B*c^4*d^2*x^2 + 10752*A*b^3*c^5*d^2*x^2
+ 6048*b^5*B*c^3*d*e*x^2 - 10752*A*b^4*c^4*d*e*x^2 - 1848*b^6*B*c^2*e^2*x^2 + 3024*A*b^5*c^3*e^2*x^2 + 4608*b^
3*B*c^5*d^2*x^3 + 580608*A*b^2*c^6*d^2*x^3 - 5184*b^4*B*c^4*d*e*x^3 + 9216*A*b^3*c^5*d*e*x^3 + 1584*b^5*B*c^3*
e^2*x^3 - 2592*A*b^4*c^4*e^2*x^3 + 454656*b^2*B*c^6*d^2*x^4 + 860160*A*b*c^7*d^2*x^4 + 4608*b^3*B*c^5*d*e*x^4
+ 909312*A*b^2*c^6*d*e*x^4 - 1408*b^4*B*c^4*e^2*x^4 + 2304*A*b^3*c^5*e^2*x^4 + 712704*b*B*c^7*d^2*x^5 + 344064
*A*c^8*d^2*x^5 + 746496*b^2*B*c^6*d*e*x^5 + 1425408*A*b*c^7*d*e*x^5 + 1280*b^3*B*c^5*e^2*x^5 + 373248*A*b^2*c^
6*e^2*x^5 + 294912*B*c^8*d^2*x^6 + 1216512*b*B*c^7*d*e*x^6 + 589824*A*c^8*d*e*x^6 + 316416*b^2*B*c^6*e^2*x^6 +
608256*A*b*c^7*e^2*x^6 + 516096*B*c^8*d*e*x^7 + 530432*b*B*c^7*e^2*x^7 + 258048*A*c^8*e^2*x^7 + 229376*B*c^8*
e^2*x^8))/(2064384*c^6) - (5*(32*b^7*B*c^2*d^2 - 64*A*b^6*c^3*d^2 - 36*b^8*B*c*d*e + 64*A*b^7*c^2*d*e + 11*b^9
*B*e^2 - 18*A*b^8*c*e^2)*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 = 0.46, size = 1292, normalized size = 3.05

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4128768*(315*(32*(B*b^7*c^2 - 2*A*b^6*c^3)*d^2 - 4*(9*B*b^8*c - 16*A*b^7*c^2)*d*e + (11*B*b^9 - 18*A*b^8*c)
*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(229376*B*c^9*e^2*x^8 + 14336*(36*B*c^9*d*e + (
37*B*b*c^8 + 18*A*c^9)*e^2)*x^7 + 3072*(96*B*c^9*d^2 + 12*(33*B*b*c^8 + 16*A*c^9)*d*e + (103*B*b^2*c^7 + 198*A
*b*c^8)*e^2)*x^6 + 256*(96*(29*B*b*c^8 + 14*A*c^9)*d^2 + 12*(243*B*b^2*c^7 + 464*A*b*c^8)*d*e + (5*B*b^3*c^6 +
1458*A*b^2*c^7)*e^2)*x^5 + 128*(96*(37*B*b^2*c^7 + 70*A*b*c^8)*d^2 + 12*(3*B*b^3*c^6 + 592*A*b^2*c^7)*d*e - (
11*B*b^4*c^5 - 18*A*b^3*c^6)*e^2)*x^4 + 144*(32*(B*b^3*c^6 + 126*A*b^2*c^7)*d^2 - 4*(9*B*b^4*c^5 - 16*A*b^3*c^
6)*d*e + (11*B*b^5*c^4 - 18*A*b^4*c^5)*e^2)*x^3 - 10080*(B*b^6*c^3 - 2*A*b^5*c^4)*d^2 + 1260*(9*B*b^7*c^2 - 16
*A*b^6*c^3)*d*e - 315*(11*B*b^8*c - 18*A*b^7*c^2)*e^2 - 168*(32*(B*b^4*c^5 - 2*A*b^3*c^6)*d^2 - 4*(9*B*b^5*c^4
- 16*A*b^4*c^5)*d*e + (11*B*b^6*c^3 - 18*A*b^5*c^4)*e^2)*x^2 + 210*(32*(B*b^5*c^4 - 2*A*b^4*c^5)*d^2 - 4*(9*B
*b^6*c^3 - 16*A*b^5*c^4)*d*e + (11*B*b^7*c^2 - 18*A*b^6*c^3)*e^2)*x)*sqrt(c*x^2 + b*x))/c^7, -1/2064384*(315*(
32*(B*b^7*c^2 - 2*A*b^6*c^3)*d^2 - 4*(9*B*b^8*c - 16*A*b^7*c^2)*d*e + (11*B*b^9 - 18*A*b^8*c)*e^2)*sqrt(-c)*ar
ctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (229376*B*c^9*e^2*x^8 + 14336*(36*B*c^9*d*e + (37*B*b*c^8 + 18*A*c^9)
*e^2)*x^7 + 3072*(96*B*c^9*d^2 + 12*(33*B*b*c^8 + 16*A*c^9)*d*e + (103*B*b^2*c^7 + 198*A*b*c^8)*e^2)*x^6 + 256
*(96*(29*B*b*c^8 + 14*A*c^9)*d^2 + 12*(243*B*b^2*c^7 + 464*A*b*c^8)*d*e + (5*B*b^3*c^6 + 1458*A*b^2*c^7)*e^2)*
x^5 + 128*(96*(37*B*b^2*c^7 + 70*A*b*c^8)*d^2 + 12*(3*B*b^3*c^6 + 592*A*b^2*c^7)*d*e - (11*B*b^4*c^5 - 18*A*b^
3*c^6)*e^2)*x^4 + 144*(32*(B*b^3*c^6 + 126*A*b^2*c^7)*d^2 - 4*(9*B*b^4*c^5 - 16*A*b^3*c^6)*d*e + (11*B*b^5*c^4
- 18*A*b^4*c^5)*e^2)*x^3 - 10080*(B*b^6*c^3 - 2*A*b^5*c^4)*d^2 + 1260*(9*B*b^7*c^2 - 16*A*b^6*c^3)*d*e - 315*
(11*B*b^8*c - 18*A*b^7*c^2)*e^2 - 168*(32*(B*b^4*c^5 - 2*A*b^3*c^6)*d^2 - 4*(9*B*b^5*c^4 - 16*A*b^4*c^5)*d*e +
(11*B*b^6*c^3 - 18*A*b^5*c^4)*e^2)*x^2 + 210*(32*(B*b^5*c^4 - 2*A*b^4*c^5)*d^2 - 4*(9*B*b^6*c^3 - 16*A*b^5*c^
4)*d*e + (11*B*b^7*c^2 - 18*A*b^6*c^3)*e^2)*x)*sqrt(c*x^2 + b*x))/c^7]

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giac [A]  time = 0.30, size = 681, normalized size = 1.61 \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 e^{2} + \frac {36 \, B c^{10} d e + 37 \, B b c^{9} e^{2} + 18 \, A c^{10} e^{2}}{c^{8}}\right )} x + \frac {3 \, {\left (96 \, B c^{10} d^{2} + 396 \, B b c^{9} d e + 192 \, A c^{10} d e + 103 \, B b^{2} c^{8} e^{2} + 198 \, A b c^{9} e^{2}\right )}}{c^{8}}\right )} x + \frac {2784 \, B b c^{9} d^{2} + 1344 \, A c^{10} d^{2} + 2916 \, B b^{2} c^{8} d e + 5568 \, A b c^{9} d e + 5 \, B b^{3} c^{7} e^{2} + 1458 \, A b^{2} c^{8} e^{2}}{c^{8}}\right )} x + \frac {3552 \, B b^{2} c^{8} d^{2} + 6720 \, A b c^{9} d^{2} + 36 \, B b^{3} c^{7} d e + 7104 \, A b^{2} c^{8} d e - 11 \, B b^{4} c^{6} e^{2} + 18 \, A b^{3} c^{7} e^{2}}{c^{8}}\right )} x + \frac {9 \, {\left (32 \, B b^{3} c^{7} d^{2} + 4032 \, A b^{2} c^{8} d^{2} - 36 \, B b^{4} c^{6} d e + 64 \, A b^{3} c^{7} d e + 11 \, B b^{5} c^{5} e^{2} - 18 \, A b^{4} c^{6} e^{2}\right )}}{c^{8}}\right )} x - \frac {21 \, {\left (32 \, B b^{4} c^{6} d^{2} - 64 \, A b^{3} c^{7} d^{2} - 36 \, B b^{5} c^{5} d e + 64 \, A b^{4} c^{6} d e + 11 \, B b^{6} c^{4} e^{2} - 18 \, A b^{5} c^{5} e^{2}\right )}}{c^{8}}\right )} x + \frac {105 \, {\left (32 \, B b^{5} c^{5} d^{2} - 64 \, A b^{4} c^{6} d^{2} - 36 \, B b^{6} c^{4} d e + 64 \, A b^{5} c^{5} d e + 11 \, B b^{7} c^{3} e^{2} - 18 \, A b^{6} c^{4} e^{2}\right )}}{c^{8}}\right )} x - \frac {315 \, {\left (32 \, B b^{6} c^{4} d^{2} - 64 \, A b^{5} c^{5} d^{2} - 36 \, B b^{7} c^{3} d e + 64 \, A b^{6} c^{4} d e + 11 \, B b^{8} c^{2} e^{2} - 18 \, A b^{7} c^{3} e^{2}\right )}}{c^{8}}\right )} - \frac {5 \, {\left (32 \, B b^{7} c^{2} d^{2} - 64 \, A b^{6} c^{3} d^{2} - 36 \, B b^{8} c d e + 64 \, A b^{7} c^{2} d e + 11 \, B b^{9} e^{2} - 18 \, A b^{8} c e^{2}\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((B*x+A)*(e*x+d)^2*(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*e^2 + (36*B*c^10*d*e + 37*B*b*c^9*e^2 + 18*A*c^1
0*e^2)/c^8)*x + 3*(96*B*c^10*d^2 + 396*B*b*c^9*d*e + 192*A*c^10*d*e + 103*B*b^2*c^8*e^2 + 198*A*b*c^9*e^2)/c^8
)*x + (2784*B*b*c^9*d^2 + 1344*A*c^10*d^2 + 2916*B*b^2*c^8*d*e + 5568*A*b*c^9*d*e + 5*B*b^3*c^7*e^2 + 1458*A*b
^2*c^8*e^2)/c^8)*x + (3552*B*b^2*c^8*d^2 + 6720*A*b*c^9*d^2 + 36*B*b^3*c^7*d*e + 7104*A*b^2*c^8*d*e - 11*B*b^4
*c^6*e^2 + 18*A*b^3*c^7*e^2)/c^8)*x + 9*(32*B*b^3*c^7*d^2 + 4032*A*b^2*c^8*d^2 - 36*B*b^4*c^6*d*e + 64*A*b^3*c
^7*d*e + 11*B*b^5*c^5*e^2 - 18*A*b^4*c^6*e^2)/c^8)*x - 21*(32*B*b^4*c^6*d^2 - 64*A*b^3*c^7*d^2 - 36*B*b^5*c^5*
d*e + 64*A*b^4*c^6*d*e + 11*B*b^6*c^4*e^2 - 18*A*b^5*c^5*e^2)/c^8)*x + 105*(32*B*b^5*c^5*d^2 - 64*A*b^4*c^6*d^
2 - 36*B*b^6*c^4*d*e + 64*A*b^5*c^5*d*e + 11*B*b^7*c^3*e^2 - 18*A*b^6*c^4*e^2)/c^8)*x - 315*(32*B*b^6*c^4*d^2
- 64*A*b^5*c^5*d^2 - 36*B*b^7*c^3*d*e + 64*A*b^6*c^4*d*e + 11*B*b^8*c^2*e^2 - 18*A*b^7*c^3*e^2)/c^8) - 5/65536
*(32*B*b^7*c^2*d^2 - 64*A*b^6*c^3*d^2 - 36*B*b^8*c*d*e + 64*A*b^7*c^2*d*e + 11*B*b^9*e^2 - 18*A*b^8*c*e^2)*log
(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(13/2)

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maple [B]  time = 0.08, size = 1227, normalized size = 2.90

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/16384*b^8/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e+45/8192*b^6/c^4*(c*x^2+b*x)^(1/2)*x*A*e
^2+45/8192*b^7/c^5*(c*x^2+b*x)^(1/2)*B*d*e-15/1024*b^5/c^4*(c*x^2+b*x)^(3/2)*B*d*e+5/96*b^3/c^2*(c*x^2+b*x)^(3
/2)*x*A*d*e+3/32*b^2/c^2*x*(c*x^2+b*x)^(5/2)*B*d*e-9/56*b/c^2*(c*x^2+b*x)^(7/2)*B*d*e+3/64*b^2/c^2*x*(c*x^2+b*
x)^(5/2)*A*e^2+1/4*x*(c*x^2+b*x)^(7/2)/c*B*d*e+1/6*A*d^2*x*(c*x^2+b*x)^(5/2)+1/7*(c*x^2+b*x)^(7/2)/c*B*d^2-1/1
2*b/c*x*(c*x^2+b*x)^(5/2)*B*d^2-1/12*b^2/c^2*(c*x^2+b*x)^(5/2)*A*d*e+5/192*b^3/c^2*(c*x^2+b*x)^(3/2)*x*B*d^2+5
/192*b^4/c^3*(c*x^2+b*x)^(3/2)*A*d*e-5/512*b^5/c^3*(c*x^2+b*x)^(1/2)*x*B*d^2-5/512*b^6/c^4*(c*x^2+b*x)^(1/2)*A
*d*e+5/1024*b^7/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d*e-5/96*A*d^2*b^2/c*(c*x^2+b*x)^(3/2)*x+5
/256*A*d^2*b^4/c^2*(c*x^2+b*x)^(1/2)*x+55/65536*B*e^2*b^9/c^(13/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+3
/128*b^3/c^3*(c*x^2+b*x)^(5/2)*A*e^2-15/2048*b^5/c^4*(c*x^2+b*x)^(3/2)*A*e^2+1/8*x*(c*x^2+b*x)^(7/2)/c*A*e^2-9
/112*b/c^2*(c*x^2+b*x)^(7/2)*A*e^2+5/2048*b^7/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d^2-11/384*B
*e^2*b^3/c^3*x*(c*x^2+b*x)^(5/2)+55/6144*B*e^2*b^5/c^4*(c*x^2+b*x)^(3/2)*x-55/16384*B*e^2*b^7/c^5*(c*x^2+b*x)^
(1/2)*x-11/144*B*e^2*b/c^2*x*(c*x^2+b*x)^(7/2)+55/12288*B*e^2*b^6/c^5*(c*x^2+b*x)^(3/2)-55/32768*B*e^2*b^8/c^6
*(c*x^2+b*x)^(1/2)+45/4096*b^6/c^4*(c*x^2+b*x)^(1/2)*x*B*d*e-1/6*b/c*x*(c*x^2+b*x)^(5/2)*A*d*e-5/256*b^5/c^3*(
c*x^2+b*x)^(1/2)*x*A*d*e-15/512*b^4/c^3*(c*x^2+b*x)^(3/2)*x*B*d*e+3/64*b^3/c^3*(c*x^2+b*x)^(5/2)*B*d*e+11/224*
B*e^2*b^2/c^3*(c*x^2+b*x)^(7/2)-45/32768*b^8/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e^2-5/1024*A
*d^2*b^6/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))-5/192*A*d^2*b^3/c^2*(c*x^2+b*x)^(3/2)+1/9*B*e^2*x^2
*(c*x^2+b*x)^(7/2)/c+45/16384*b^7/c^5*(c*x^2+b*x)^(1/2)*A*e^2-15/1024*b^4/c^3*(c*x^2+b*x)^(3/2)*x*A*e^2-1/24*b
^2/c^2*(c*x^2+b*x)^(5/2)*B*d^2+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*B*d^2-5/1024*b^6/c^4*(c*x^2+b*x)^(1/2)*B*d^2-11
/768*B*e^2*b^4/c^4*(c*x^2+b*x)^(5/2)+2/7*(c*x^2+b*x)^(7/2)/c*A*d*e+5/512*A*d^2*b^5/c^3*(c*x^2+b*x)^(1/2)+1/12*
A*d^2/c*(c*x^2+b*x)^(5/2)*b

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maxima [B]  time = 0.60, size = 944, normalized size = 2.23

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

1/9*(c*x^2 + b*x)^(7/2)*B*e^2*x^2/c + 1/6*(c*x^2 + b*x)^(5/2)*A*d^2*x + 5/256*sqrt(c*x^2 + b*x)*A*b^4*d^2*x/c^
2 - 5/96*(c*x^2 + b*x)^(3/2)*A*b^2*d^2*x/c - 55/16384*sqrt(c*x^2 + b*x)*B*b^7*e^2*x/c^5 + 55/6144*(c*x^2 + b*x
)^(3/2)*B*b^5*e^2*x/c^4 - 11/384*(c*x^2 + b*x)^(5/2)*B*b^3*e^2*x/c^3 - 11/144*(c*x^2 + b*x)^(7/2)*B*b*e^2*x/c^
2 - 5/1024*A*b^6*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 55/65536*B*b^9*e^2*log(2*c*x + b +
2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(13/2) + 5/512*sqrt(c*x^2 + b*x)*A*b^5*d^2/c^3 - 5/192*(c*x^2 + b*x)^(3/2)*A*b
^3*d^2/c^2 + 1/12*(c*x^2 + b*x)^(5/2)*A*b*d^2/c - 55/32768*sqrt(c*x^2 + b*x)*B*b^8*e^2/c^6 + 55/12288*(c*x^2 +
b*x)^(3/2)*B*b^6*e^2/c^5 - 11/768*(c*x^2 + b*x)^(5/2)*B*b^4*e^2/c^4 + 11/224*(c*x^2 + b*x)^(7/2)*B*b^2*e^2/c^
3 + 45/8192*(2*B*d*e + A*e^2)*sqrt(c*x^2 + b*x)*b^6*x/c^4 - 15/1024*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(3/2)*b^4*
x/c^3 - 5/512*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^5*x/c^3 + 3/64*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(5/2)*b^2*x
/c^2 + 5/192*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2)*b^3*x/c^2 + 1/8*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(7/2)*x/c -
1/12*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(5/2)*b*x/c - 45/32768*(2*B*d*e + A*e^2)*b^8*log(2*c*x + b + 2*sqrt(c*x^
2 + b*x)*sqrt(c))/c^(11/2) + 5/2048*(B*d^2 + 2*A*d*e)*b^7*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2)
+ 45/16384*(2*B*d*e + A*e^2)*sqrt(c*x^2 + b*x)*b^7/c^5 - 15/2048*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(3/2)*b^5/c^
4 - 5/1024*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^6/c^4 + 3/128*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(5/2)*b^3/c^3 +
5/384*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2)*b^4/c^3 - 9/112*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(7/2)*b/c^2 - 1/2
4*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(5/2)*b^2/c^2 + 1/7*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(7/2)/c

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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