3.1565 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=446 \[ \frac{3 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{65536 c^6}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{2560 c^4}+\frac{\left (a+b x+c x^2\right )^{7/2} \left (14 c e x \left (-4 c e (9 a e+2 b d)+11 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (160 a e+17 b d)+4 b c e^2 (97 a e+90 b d)-99 b^3 e^3+128 c^3 d^3\right )}{6720 c^3}-\frac{3 e \left (b^2-4 a c\right )^4 \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{131072 c^{13/2}}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{30 c} \]

[Out]

(3*(b^2 - 4*a*c)^3*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(6553
6*c^6) - ((b^2 - 4*a*c)^2*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/
2))/(8192*c^5) + ((b^2 - 4*a*c)*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^
2)^(5/2))/(2560*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(30*c) + ((d + e*x)^3*(a + b*x + c*
x^2)^(7/2))/5 + ((128*c^3*d^3 - 99*b^3*e^3 + 4*b*c*e^2*(90*b*d + 97*a*e) - 8*c^2*d*e*(17*b*d + 160*a*e) + 14*c
*e*(8*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(2*b*d + 9*a*e))*x)*(a + b*x + c*x^2)^(7/2))/(6720*c^3) - (3*(b^2 - 4*a*c)^
4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
131072*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.62374, antiderivative size = 446, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {832, 779, 612, 621, 206} \[ \frac{3 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{65536 c^6}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right )}{2560 c^4}+\frac{\left (a+b x+c x^2\right )^{7/2} \left (14 c e x \left (-4 c e (9 a e+2 b d)+11 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (160 a e+17 b d)+4 b c e^2 (97 a e+90 b d)-99 b^3 e^3+128 c^3 d^3\right )}{6720 c^3}-\frac{3 e \left (b^2-4 a c\right )^4 \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{131072 c^{13/2}}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{30 c} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(3*(b^2 - 4*a*c)^3*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(6553
6*c^6) - ((b^2 - 4*a*c)^2*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/
2))/(8192*c^5) + ((b^2 - 4*a*c)*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^
2)^(5/2))/(2560*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(30*c) + ((d + e*x)^3*(a + b*x + c*
x^2)^(7/2))/5 + ((128*c^3*d^3 - 99*b^3*e^3 + 4*b*c*e^2*(90*b*d + 97*a*e) - 8*c^2*d*e*(17*b*d + 160*a*e) + 14*c
*e*(8*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(2*b*d + 9*a*e))*x)*(a + b*x + c*x^2)^(7/2))/(6720*c^3) - (3*(b^2 - 4*a*c)^
4*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(10*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(
131072*c^(13/2))

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])

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 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 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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 (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\int (d+e x)^2 (3 c (b d-2 a e)+3 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{5/2} \, dx}{10 c}\\ &=\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\int (d+e x) \left (\frac{3}{2} c \left (7 b^2 d e-44 a c d e+4 b \left (c d^2+a e^2\right )\right )+\frac{3}{2} c \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}+\frac{\left (3 \left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{640 c^3}\\ &=\frac{\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac{\left (\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}+\frac{\left (3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{16384 c^5}\\ &=\frac{3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{65536 c^6}-\frac{\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac{\left (3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{131072 c^6}\\ &=\frac{3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{65536 c^6}-\frac{\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac{\left (3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{65536 c^6}\\ &=\frac{3 \left (b^2-4 a c\right )^3 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{65536 c^6}-\frac{\left (b^2-4 a c\right )^2 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{2560 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{30 c}+\frac{1}{5} (d+e x)^3 \left (a+b x+c x^2\right )^{7/2}+\frac{\left (128 c^3 d^3-99 b^3 e^3+4 b c e^2 (90 b d+97 a e)-8 c^2 d e (17 b d+160 a e)+14 c e \left (8 c^2 d^2+11 b^2 e^2-4 c e (2 b d+9 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{6720 c^3}-\frac{3 \left (b^2-4 a c\right )^4 e \left (40 c^2 d^2+11 b^2 e^2-4 c e (10 b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{131072 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 1.11821, size = 345, normalized size = 0.77 \[ \frac{1}{10} \left (-\frac{e \left (b^2-4 a c\right ) \left (-4 c e (a e+10 b d)+11 b^2 e^2+40 c^2 d^2\right ) \left (15 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )-40 b^3 c x+15 b^4\right )\right )}{65536 c^{13/2}}+\frac{(a+x (b+c x))^{7/2} \left (-8 c^2 e (a e (160 d+63 e x)+b d (17 d+14 e x))+2 b c e^2 (194 a e+180 b d+77 b e x)-99 b^3 e^3+16 c^3 d^2 (8 d+7 e x)\right )}{672 c^3}+2 (d+e x)^3 (a+x (b+c x))^{7/2}+\frac{(d+e x)^2 (a+x (b+c x))^{7/2} (2 c d-b e)}{3 c}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(((2*c*d - b*e)*(d + e*x)^2*(a + x*(b + c*x))^(7/2))/(3*c) + 2*(d + e*x)^3*(a + x*(b + c*x))^(7/2) + ((a + x*(
b + c*x))^(7/2)*(-99*b^3*e^3 + 16*c^3*d^2*(8*d + 7*e*x) + 2*b*c*e^2*(180*b*d + 194*a*e + 77*b*e*x) - 8*c^2*e*(
b*d*(17*d + 14*e*x) + a*e*(160*d + 63*e*x))))/(672*c^3) - ((b^2 - 4*a*c)*e*(40*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(1
0*b*d + a*e))*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2)
+ 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)) + 15*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*
c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(65536*c^(13/2)))/10

________________________________________________________________________________________

Maple [B]  time = 0.02, size = 2387, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x)

[Out]

-51/4096/c^4*e^3*b^6*(c*x^2+b*x+a)^(1/2)*x*a-13/256/c^2*e^3*b^2*a^2*(c*x^2+b*x+a)^(3/2)*x-21/256/c^2*e^3*b^2*a
^3*(c*x^2+b*x+a)^(1/2)*x+23/1024/c^3*e^3*b^4*(c*x^2+b*x+a)^(3/2)*x*a-3/80/c^2*e^3*b^2*a*x*(c*x^2+b*x+a)^(5/2)-
45/512*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d^2*e+15/1024*b^6/c^(7/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^2*e+15/64*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3*d^2*e+15/
64*b/c^(3/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2+45/512*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*a^2*d*e^2-15/1024*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e^2-15/64*b^
3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3*d*e^2+15/4096*b^6/c^3*(c*x^2+b*x+a)^(1/2)*x*d^2*e+45
/512*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a^2*d^2*e-45/2048*b^5/c^3*(c*x^2+b*x+a)^(1/2)*a*d^2*e+1/32*b^2/c*x*(c*x^2+b*x
+a)^(5/2)*d^2*e-15/128*a^3/c*(c*x^2+b*x+a)^(1/2)*b*d^2*e-1/16*a/c*(c*x^2+b*x+a)^(5/2)*b*d^2*e-5/64*a^2/c*(c*x^
2+b*x+a)^(3/2)*b*d^2*e-5/512*b^4/c^2*(c*x^2+b*x+a)^(3/2)*x*d^2*e+5/128*b^3/c^2*(c*x^2+b*x+a)^(3/2)*a*d^2*e+5/5
12*b^5/c^3*(c*x^2+b*x+a)^(3/2)*x*d*e^2-5/128*b^4/c^3*(c*x^2+b*x+a)^(3/2)*a*d*e^2+5/64*b^2/c^2*a^2*(c*x^2+b*x+a
)^(3/2)*d*e^2-1/32*b^3/c^2*x*(c*x^2+b*x+a)^(5/2)*d*e^2-15/4096*b^7/c^4*(c*x^2+b*x+a)^(1/2)*x*d*e^2-45/512*b^4/
c^3*(c*x^2+b*x+a)^(1/2)*a^2*d*e^2+45/2048*b^6/c^4*(c*x^2+b*x+a)^(1/2)*a*d*e^2-1/12*b/c*x*(c*x^2+b*x+a)^(7/2)*d
*e^2+15/128*b^2/c^2*a^3*(c*x^2+b*x+a)^(1/2)*d*e^2+1/16*b^2/c^2*a*(c*x^2+b*x+a)^(5/2)*d*e^2+27/512/c^3*e^3*b^4*
(c*x^2+b*x+a)^(1/2)*x*a^2+3/4*x*(c*x^2+b*x+a)^(7/2)*d^2*e+2/3*x^2*(c*x^2+b*x+a)^(7/2)*d*e^2-33/2240/c^3*e^3*b^
3*(c*x^2+b*x+a)^(7/2)+11/2560/c^4*e^3*b^5*(c*x^2+b*x+a)^(5/2)-11/8192/c^5*e^3*b^7*(c*x^2+b*x+a)^(3/2)+33/65536
/c^6*e^3*b^9*(c*x^2+b*x+a)^(1/2)+3/128/c^(3/2)*e^3*a^5*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-33/131072/c
^(13/2)*e^3*b^10*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2/7*(c*x^2+b*x+a)^(7/2)*d^3+1/5*e^3*x^3*(c*x^2+b*
x+a)^(7/2)+27/1024/c^4*e^3*b^5*(c*x^2+b*x+a)^(1/2)*a^2-1/30/c*e^3*b*x^2*(c*x^2+b*x+a)^(7/2)+97/1680/c^2*e^3*b*
a*(c*x^2+b*x+a)^(7/2)+23/2048/c^4*e^3*b^5*(c*x^2+b*x+a)^(3/2)*a+33/32768/c^5*e^3*b^8*(c*x^2+b*x+a)^(1/2)*x+11/
1280/c^3*e^3*b^4*x*(c*x^2+b*x+a)^(5/2)+1/64/c*e^3*a^3*(c*x^2+b*x+a)^(3/2)*x-11/4096/c^4*e^3*b^6*(c*x^2+b*x+a)^
(3/2)*x+1/128/c^2*e^3*a^3*(c*x^2+b*x+a)^(3/2)*b-51/8192/c^5*e^3*b^7*(c*x^2+b*x+a)^(1/2)*a+11/480/c^2*e^3*b^2*x
*(c*x^2+b*x+a)^(7/2)-21/512/c^3*e^3*b^3*a^3*(c*x^2+b*x+a)^(1/2)-3/160/c^3*e^3*b^3*a*(c*x^2+b*x+a)^(5/2)-13/512
/c^3*e^3*b^3*a^2*(c*x^2+b*x+a)^(3/2)+1/64*b^3/c^2*(c*x^2+b*x+a)^(5/2)*d^2*e-5/1024*b^5/c^3*(c*x^2+b*x+a)^(3/2)
*d^2*e+15/8192*b^7/c^4*(c*x^2+b*x+a)^(1/2)*d^2*e-3/56*b/c*(c*x^2+b*x+a)^(7/2)*d^2*e-4/21*a/c*(c*x^2+b*x+a)^(7/
2)*d*e^2+3/56*b^2/c^2*(c*x^2+b*x+a)^(7/2)*d*e^2-1/64*b^4/c^3*(c*x^2+b*x+a)^(5/2)*d*e^2+5/1024*b^6/c^4*(c*x^2+b
*x+a)^(3/2)*d*e^2-15/8192*b^8/c^5*(c*x^2+b*x+a)^(1/2)*d*e^2+3/128/c*e^3*a^4*(c*x^2+b*x+a)^(1/2)*x-3/40/c*e^3*a
*x*(c*x^2+b*x+a)^(7/2)+3/256/c^2*e^3*a^4*(c*x^2+b*x+a)^(1/2)*b+1/80/c*e^3*a^2*x*(c*x^2+b*x+a)^(5/2)+1/160/c^2*
e^3*a^2*(c*x^2+b*x+a)^(5/2)*b+15/16384*b^9/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2+135/3276
8/c^(11/2)*e^3*b^8*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+75/1024/c^(7/2)*e^3*b^4*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*a^3-45/512/c^(5/2)*e^3*b^2*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-105/4096/c^(
9/2)*e^3*b^6*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-1/8*a*x*(c*x^2+b*x+a)^(5/2)*d^2*e-5/32*a^2*(c*x^2
+b*x+a)^(3/2)*x*d^2*e-15/64*a^3*(c*x^2+b*x+a)^(1/2)*x*d^2*e-15/64*a^4/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*d^2*e-15/16384*b^8/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e-45/1024*b^4/c^2*(c*x^
2+b*x+a)^(1/2)*x*a*d^2*e+5/64*b^2/c*(c*x^2+b*x+a)^(3/2)*x*a*d^2*e+45/256*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a^2*d^2*e
+15/64*b/c*a^3*(c*x^2+b*x+a)^(1/2)*x*d*e^2-5/64*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x*a*d*e^2-45/256*b^3/c^2*(c*x^2+b*
x+a)^(1/2)*x*a^2*d*e^2+45/1024*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x*a*d*e^2+5/32*b/c*a^2*(c*x^2+b*x+a)^(3/2)*x*d*e^2+
1/8*b/c*a*x*(c*x^2+b*x+a)^(5/2)*d*e^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((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.75273, size = 5612, normalized size = 12.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/27525120*(315*(40*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^3*b^2*c^5 + 256*a^4*c^6)*d^2*e - 40*(b^
9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 256*a^3*b^3*c^4 + 256*a^4*b*c^5)*d*e^2 + (11*b^10 - 180*a*b^8*c + 1120*a
^2*b^6*c^2 - 3200*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 -
4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(1376256*c^10*e^3*x^9 + 1966080*a^3*c^7*d^3 + 229376
*(20*c^10*d*e^2 + 17*b*c^9*e^3)*x^8 + 14336*(360*c^10*d^2*e + 920*b*c^9*d*e^2 + (251*b^2*c^8 + 252*a*c^9)*e^3)
*x^7 + 1024*(1920*c^10*d^3 + 14760*b*c^9*d^2*e + 40*(303*b^2*c^8 + 304*a*c^9)*d*e^2 + (1035*b^3*c^7 + 6268*a*b
*c^8)*e^3)*x^6 + 256*(23040*b*c^9*d^3 + 120*(475*b^2*c^8 + 476*a*c^9)*d^2*e + 40*(367*b^3*c^7 + 2220*a*b*c^8)*
d*e^2 - (5*b^4*c^6 - 10200*a*b^2*c^7 - 10416*a^2*c^8)*e^3)*x^5 + 128*(46080*(b^2*c^8 + a*c^9)*d^3 + 120*(299*b
^3*c^7 + 1804*a*b*c^8)*d^2*e - 40*(b^4*c^6 - 1884*a*b^2*c^7 - 1920*a^2*c^8)*d*e^2 + (11*b^5*c^5 - 120*a*b^3*c^
6 + 13680*a^2*b*c^7)*e^3)*x^4 + 120*(105*b^7*c^3 - 1540*a*b^5*c^4 + 8176*a^2*b^3*c^5 - 17856*a^3*b*c^6)*d^2*e
- 40*(315*b^8*c^2 - 4620*a*b^6*c^3 + 24528*a^2*b^4*c^4 - 53568*a^3*b^2*c^5 + 32768*a^4*c^6)*d*e^2 + (3465*b^9*
c - 52080*a*b^7*c^2 + 288288*a^2*b^5*c^3 - 687360*a^3*b^3*c^4 + 574720*a^4*b*c^5)*e^3 + 16*(122880*(b^3*c^7 +
6*a*b*c^8)*d^3 - 120*(3*b^4*c^6 - 6520*a*b^2*c^7 - 6608*a^2*c^8)*d^2*e + 40*(9*b^5*c^5 - 104*a*b^3*c^6 + 10896
*a^2*b*c^7)*d*e^2 - (99*b^6*c^4 - 1180*a*b^4*c^5 + 4560*a^2*b^2*c^6 - 6720*a^3*c^7)*e^3)*x^3 + 8*(737280*(a*b^
2*c^7 + a^2*c^8)*d^3 + 120*(7*b^5*c^5 - 88*a*b^3*c^6 + 10608*a^2*b*c^7)*d^2*e - 40*(21*b^6*c^4 - 264*a*b^4*c^5
+ 1104*a^2*b^2*c^6 - 2048*a^3*c^7)*d*e^2 + (231*b^7*c^3 - 2988*a*b^5*c^4 + 13200*a^2*b^3*c^5 - 20800*a^3*b*c^
6)*e^3)*x^2 + 2*(2949120*a^2*b*c^7*d^3 - 120*(35*b^6*c^4 - 476*a*b^4*c^5 + 2256*a^2*b^2*c^6 - 6720*a^3*c^7)*d^
2*e + 40*(105*b^7*c^3 - 1428*a*b^5*c^4 + 6768*a^2*b^3*c^5 - 11968*a^3*b*c^6)*d*e^2 - (1155*b^8*c^2 - 16128*a*b
^6*c^3 + 80160*a^2*b^4*c^4 - 158720*a^3*b^2*c^5 + 80640*a^4*c^6)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, 1/1376256
0*(315*(40*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^3*b^2*c^5 + 256*a^4*c^6)*d^2*e - 40*(b^9*c - 16*a*
b^7*c^2 + 96*a^2*b^5*c^3 - 256*a^3*b^3*c^4 + 256*a^4*b*c^5)*d*e^2 + (11*b^10 - 180*a*b^8*c + 1120*a^2*b^6*c^2
- 3200*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x +
b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(1376256*c^10*e^3*x^9 + 1966080*a^3*c^7*d^3 + 229376*(20*c^10*d*e^2 +
17*b*c^9*e^3)*x^8 + 14336*(360*c^10*d^2*e + 920*b*c^9*d*e^2 + (251*b^2*c^8 + 252*a*c^9)*e^3)*x^7 + 1024*(1920
*c^10*d^3 + 14760*b*c^9*d^2*e + 40*(303*b^2*c^8 + 304*a*c^9)*d*e^2 + (1035*b^3*c^7 + 6268*a*b*c^8)*e^3)*x^6 +
256*(23040*b*c^9*d^3 + 120*(475*b^2*c^8 + 476*a*c^9)*d^2*e + 40*(367*b^3*c^7 + 2220*a*b*c^8)*d*e^2 - (5*b^4*c^
6 - 10200*a*b^2*c^7 - 10416*a^2*c^8)*e^3)*x^5 + 128*(46080*(b^2*c^8 + a*c^9)*d^3 + 120*(299*b^3*c^7 + 1804*a*b
*c^8)*d^2*e - 40*(b^4*c^6 - 1884*a*b^2*c^7 - 1920*a^2*c^8)*d*e^2 + (11*b^5*c^5 - 120*a*b^3*c^6 + 13680*a^2*b*c
^7)*e^3)*x^4 + 120*(105*b^7*c^3 - 1540*a*b^5*c^4 + 8176*a^2*b^3*c^5 - 17856*a^3*b*c^6)*d^2*e - 40*(315*b^8*c^2
- 4620*a*b^6*c^3 + 24528*a^2*b^4*c^4 - 53568*a^3*b^2*c^5 + 32768*a^4*c^6)*d*e^2 + (3465*b^9*c - 52080*a*b^7*c
^2 + 288288*a^2*b^5*c^3 - 687360*a^3*b^3*c^4 + 574720*a^4*b*c^5)*e^3 + 16*(122880*(b^3*c^7 + 6*a*b*c^8)*d^3 -
120*(3*b^4*c^6 - 6520*a*b^2*c^7 - 6608*a^2*c^8)*d^2*e + 40*(9*b^5*c^5 - 104*a*b^3*c^6 + 10896*a^2*b*c^7)*d*e^2
- (99*b^6*c^4 - 1180*a*b^4*c^5 + 4560*a^2*b^2*c^6 - 6720*a^3*c^7)*e^3)*x^3 + 8*(737280*(a*b^2*c^7 + a^2*c^8)*
d^3 + 120*(7*b^5*c^5 - 88*a*b^3*c^6 + 10608*a^2*b*c^7)*d^2*e - 40*(21*b^6*c^4 - 264*a*b^4*c^5 + 1104*a^2*b^2*c
^6 - 2048*a^3*c^7)*d*e^2 + (231*b^7*c^3 - 2988*a*b^5*c^4 + 13200*a^2*b^3*c^5 - 20800*a^3*b*c^6)*e^3)*x^2 + 2*(
2949120*a^2*b*c^7*d^3 - 120*(35*b^6*c^4 - 476*a*b^4*c^5 + 2256*a^2*b^2*c^6 - 6720*a^3*c^7)*d^2*e + 40*(105*b^7
*c^3 - 1428*a*b^5*c^4 + 6768*a^2*b^3*c^5 - 11968*a^3*b*c^6)*d*e^2 - (1155*b^8*c^2 - 16128*a*b^6*c^3 + 80160*a^
2*b^4*c^4 - 158720*a^3*b^2*c^5 + 80640*a^4*c^6)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^7]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**3*(a + b*x + c*x**2)**(5/2), x)

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Giac [B]  time = 1.39301, size = 1758, normalized size = 3.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/6881280*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*(6*c^3*x*e^3 + (20*c^12*d*e^2 + 17*b*c^11*e^3)/c^9)*
x + (360*c^12*d^2*e + 920*b*c^11*d*e^2 + 251*b^2*c^10*e^3 + 252*a*c^11*e^3)/c^9)*x + (1920*c^12*d^3 + 14760*b*
c^11*d^2*e + 12120*b^2*c^10*d*e^2 + 12160*a*c^11*d*e^2 + 1035*b^3*c^9*e^3 + 6268*a*b*c^10*e^3)/c^9)*x + (23040
*b*c^11*d^3 + 57000*b^2*c^10*d^2*e + 57120*a*c^11*d^2*e + 14680*b^3*c^9*d*e^2 + 88800*a*b*c^10*d*e^2 - 5*b^4*c
^8*e^3 + 10200*a*b^2*c^9*e^3 + 10416*a^2*c^10*e^3)/c^9)*x + (46080*b^2*c^10*d^3 + 46080*a*c^11*d^3 + 35880*b^3
*c^9*d^2*e + 216480*a*b*c^10*d^2*e - 40*b^4*c^8*d*e^2 + 75360*a*b^2*c^9*d*e^2 + 76800*a^2*c^10*d*e^2 + 11*b^5*
c^7*e^3 - 120*a*b^3*c^8*e^3 + 13680*a^2*b*c^9*e^3)/c^9)*x + (122880*b^3*c^9*d^3 + 737280*a*b*c^10*d^3 - 360*b^
4*c^8*d^2*e + 782400*a*b^2*c^9*d^2*e + 792960*a^2*c^10*d^2*e + 360*b^5*c^7*d*e^2 - 4160*a*b^3*c^8*d*e^2 + 4358
40*a^2*b*c^9*d*e^2 - 99*b^6*c^6*e^3 + 1180*a*b^4*c^7*e^3 - 4560*a^2*b^2*c^8*e^3 + 6720*a^3*c^9*e^3)/c^9)*x + (
737280*a*b^2*c^9*d^3 + 737280*a^2*c^10*d^3 + 840*b^5*c^7*d^2*e - 10560*a*b^3*c^8*d^2*e + 1272960*a^2*b*c^9*d^2
*e - 840*b^6*c^6*d*e^2 + 10560*a*b^4*c^7*d*e^2 - 44160*a^2*b^2*c^8*d*e^2 + 81920*a^3*c^9*d*e^2 + 231*b^7*c^5*e
^3 - 2988*a*b^5*c^6*e^3 + 13200*a^2*b^3*c^7*e^3 - 20800*a^3*b*c^8*e^3)/c^9)*x + (2949120*a^2*b*c^9*d^3 - 4200*
b^6*c^6*d^2*e + 57120*a*b^4*c^7*d^2*e - 270720*a^2*b^2*c^8*d^2*e + 806400*a^3*c^9*d^2*e + 4200*b^7*c^5*d*e^2 -
57120*a*b^5*c^6*d*e^2 + 270720*a^2*b^3*c^7*d*e^2 - 478720*a^3*b*c^8*d*e^2 - 1155*b^8*c^4*e^3 + 16128*a*b^6*c^
5*e^3 - 80160*a^2*b^4*c^6*e^3 + 158720*a^3*b^2*c^7*e^3 - 80640*a^4*c^8*e^3)/c^9)*x + (1966080*a^3*c^9*d^3 + 12
600*b^7*c^5*d^2*e - 184800*a*b^5*c^6*d^2*e + 981120*a^2*b^3*c^7*d^2*e - 2142720*a^3*b*c^8*d^2*e - 12600*b^8*c^
4*d*e^2 + 184800*a*b^6*c^5*d*e^2 - 981120*a^2*b^4*c^6*d*e^2 + 2142720*a^3*b^2*c^7*d*e^2 - 1310720*a^4*c^8*d*e^
2 + 3465*b^9*c^3*e^3 - 52080*a*b^7*c^4*e^3 + 288288*a^2*b^5*c^5*e^3 - 687360*a^3*b^3*c^6*e^3 + 574720*a^4*b*c^
7*e^3)/c^9) + 3/131072*(40*b^8*c^2*d^2*e - 640*a*b^6*c^3*d^2*e + 3840*a^2*b^4*c^4*d^2*e - 10240*a^3*b^2*c^5*d^
2*e + 10240*a^4*c^6*d^2*e - 40*b^9*c*d*e^2 + 640*a*b^7*c^2*d*e^2 - 3840*a^2*b^5*c^3*d*e^2 + 10240*a^3*b^3*c^4*
d*e^2 - 10240*a^4*b*c^5*d*e^2 + 11*b^10*e^3 - 180*a*b^8*c*e^3 + 1120*a^2*b^6*c^2*e^3 - 3200*a^3*b^4*c^3*e^3 +
3840*a^4*b^2*c^4*e^3 - 1024*a^5*c^5*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)