3.1557 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=379 \[ -\frac{3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac{3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \]

[Out]

(-3*(b^2 - 4*a*c)^2*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*
c^5) + ((b^2 - 4*a*c)*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(1
024*c^4) + (3*(2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(56*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(5/2)
)/4 + ((96*c^3*d^3 - 63*b^3*e^3 + 4*b*c*e^2*(56*b*d + 61*a*e) - 8*c^2*d*e*(13*b*d + 96*a*e) + 10*c*e*(8*c^2*d^
2 + 9*b^2*e^2 - 4*c*e*(2*b*d + 7*a*e))*x)*(a + b*x + c*x^2)^(5/2))/(2240*c^3) + (3*(b^2 - 4*a*c)^3*e*(32*c^2*d
^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16384*c^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.513945, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{3 e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{8192 c^5}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{1024 c^4}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (10 c e x \left (-4 c e (7 a e+2 b d)+9 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (96 a e+13 b d)+4 b c e^2 (61 a e+56 b d)-63 b^3 e^3+96 c^3 d^3\right )}{2240 c^3}+\frac{3 e \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{3 (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{56 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)^2*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*
c^5) + ((b^2 - 4*a*c)*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(1
024*c^4) + (3*(2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(56*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(5/2)
)/4 + ((96*c^3*d^3 - 63*b^3*e^3 + 4*b*c*e^2*(56*b*d + 61*a*e) - 8*c^2*d*e*(13*b*d + 96*a*e) + 10*c*e*(8*c^2*d^
2 + 9*b^2*e^2 - 4*c*e*(2*b*d + 7*a*e))*x)*(a + b*x + c*x^2)^(5/2))/(2240*c^3) + (3*(b^2 - 4*a*c)^3*e*(32*c^2*d
^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16384*c^(11/2))

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 )^{3/2} \, dx &=\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\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 )^{3/2} \, dx}{8 c}\\ &=\frac{3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{\int (d+e x) \left (\frac{3}{2} c \left (5 b^2 d e-36 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+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{56 c^2}\\ &=\frac{3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac{\left (\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{128 c^3}\\ &=\frac{\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac{3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}-\frac{\left (3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{2048 c^4}\\ &=-\frac{3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac{3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac{\left (3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16384 c^5}\\ &=-\frac{3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac{3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac{\left (3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 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 )}{8192 c^5}\\ &=-\frac{3 \left (b^2-4 a c\right )^2 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^5}+\frac{\left (b^2-4 a c\right ) e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{1024 c^4}+\frac{3 (2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{56 c}+\frac{1}{4} (d+e x)^3 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (96 c^3 d^3-63 b^3 e^3+4 b c e^2 (56 b d+61 a e)-8 c^2 d e (13 b d+96 a e)+10 c e \left (8 c^2 d^2+9 b^2 e^2-4 c e (2 b d+7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{2240 c^3}+\frac{3 \left (b^2-4 a c\right )^3 e \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.724127, size = 297, normalized size = 0.78 \[ \frac{1}{8} \left (\frac{e \left (b^2-4 a c\right ) \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{2048 c^{11/2}}+\frac{(a+x (b+c x))^{5/2} \left (-8 c^2 e (a e (96 d+35 e x)+b d (13 d+10 e x))+2 b c e^2 (122 a e+112 b d+45 b e x)-63 b^3 e^3+16 c^3 d^2 (6 d+5 e x)\right )}{280 c^3}+2 (d+e x)^3 (a+x (b+c x))^{5/2}+\frac{3 (d+e x)^2 (a+x (b+c x))^{5/2} (2 c d-b e)}{7 c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(2*c*d - b*e)*(d + e*x)^2*(a + x*(b + c*x))^(5/2))/(7*c) + 2*(d + e*x)^3*(a + x*(b + c*x))^(5/2) + ((a + x
*(b + c*x))^(5/2)*(-63*b^3*e^3 + 16*c^3*d^2*(6*d + 5*e*x) + 2*b*c*e^2*(112*b*d + 122*a*e + 45*b*e*x) - 8*c^2*e
*(b*d*(13*d + 10*e*x) + a*e*(96*d + 35*e*x))))/(280*c^3) + ((b^2 - 4*a*c)*e*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8
*b*d + a*e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 -
4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2048*c^(11/2)))/8

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Maple [B]  time = 0.019, size = 1607, normalized size = 4.2 \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)^(3/2),x)

[Out]

3/16*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a*d^2*e+1/4*b/c*a*x*(c*x^2+b*x+a)^(3/2)*d*e^2+3/8*b/c*a^2*(c*x^2+b*x+a)^(1/2)
*x*d*e^2-3/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a*d*e^2-3/128*b^4/c^2*(c*x^2+b*x+a)^(1/2)*x*d^2*e+3/32*b^3/c^2*(c*
x^2+b*x+a)^(1/2)*a*d^2*e+3/16*b^2/c^2*a^2*(c*x^2+b*x+a)^(1/2)*d*e^2-1/7*b/c*x*(c*x^2+b*x+a)^(5/2)*d*e^2-1/16*b
^3/c^2*x*(c*x^2+b*x+a)^(3/2)*d*e^2+1/8*b^2/c^2*a*(c*x^2+b*x+a)^(3/2)*d*e^2+3/128*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x
*d*e^2-3/32*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a*d*e^2-33/256/c^2*e^3*b^2*a^2*(c*x^2+b*x+a)^(1/2)*x-5/64/c^2*e^3*b^2*
a*x*(c*x^2+b*x+a)^(3/2)+57/1024/c^3*e^3*b^4*(c*x^2+b*x+a)^(1/2)*x*a-1/8*a/c*(c*x^2+b*x+a)^(3/2)*b*d^2*e-3/16*a
^2/c*(c*x^2+b*x+a)^(1/2)*b*d^2*e+9/32*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d^2*e-9/128*
b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^2*e+9/128*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*a*d*e^2+3/8*b/c^(3/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2-9/32*b^3/c^(5/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*e^2+1/16*b^2/c*x*(c*x^2+b*x+a)^(3/2)*d^2*e-27/8192/c^5*e^3*b^
7*(c*x^2+b*x+a)^(1/2)+3/64/c^(3/2)*e^3*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+27/16384/c^(11/2)*e^3*b
^8*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+x*(c*x^2+b*x+a)^(5/2)*d^2*e+6/7*x^2*(c*x^2+b*x+a)^(5/2)*d*e^2-9
/320/c^3*e^3*b^3*(c*x^2+b*x+a)^(5/2)+9/1024/c^4*e^3*b^5*(c*x^2+b*x+a)^(3/2)+2/5*(c*x^2+b*x+a)^(5/2)*d^3+1/4*e^
3*x^3*(c*x^2+b*x+a)^(5/2)+61/560/c^2*e^3*b*a*(c*x^2+b*x+a)^(5/2)-3/56/c*e^3*b*x^2*(c*x^2+b*x+a)^(5/2)+1/64/c^2
*e^3*a^2*(c*x^2+b*x+a)^(3/2)*b+3/64/c*e^3*a^3*(c*x^2+b*x+a)^(1/2)*x+3/128/c^2*e^3*a^3*(c*x^2+b*x+a)^(1/2)*b-1/
8/c*e^3*a*x*(c*x^2+b*x+a)^(5/2)-33/512/c^3*e^3*b^3*a^2*(c*x^2+b*x+a)^(1/2)-1/10*b/c*(c*x^2+b*x+a)^(5/2)*d^2*e+
1/32*b^3/c^2*(c*x^2+b*x+a)^(3/2)*d^2*e-3/256*b^5/c^3*(c*x^2+b*x+a)^(1/2)*d^2*e-1/32*b^4/c^3*(c*x^2+b*x+a)^(3/2
)*d*e^2+3/256*b^6/c^4*(c*x^2+b*x+a)^(1/2)*d*e^2-12/35*a/c*(c*x^2+b*x+a)^(5/2)*d*e^2+1/10*b^2/c^2*(c*x^2+b*x+a)
^(5/2)*d*e^2+1/32/c*e^3*a^2*x*(c*x^2+b*x+a)^(3/2)+9/224/c^2*e^3*b^2*x*(c*x^2+b*x+a)^(5/2)+9/512/c^3*e^3*b^4*x*
(c*x^2+b*x+a)^(3/2)-27/4096/c^4*e^3*b^6*(c*x^2+b*x+a)^(1/2)*x+57/2048/c^4*e^3*b^5*(c*x^2+b*x+a)^(1/2)*a-5/128/
c^3*e^3*b^3*a*(c*x^2+b*x+a)^(3/2)-3/512*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2+45/512/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^2-21/1024/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-9/64/c^(5/2)*e^3*b^2*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/8*a^2*(c*x^2+b*x
+a)^(1/2)*x*d^2*e-1/4*a*x*(c*x^2+b*x+a)^(3/2)*d^2*e+3/512*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))*d^2*e-3/8*a^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.7993, size = 3675, normalized size = 9.7 \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)^(3/2),x, algorithm="fricas")

[Out]

[1/1146880*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e - 32*(b^7*c - 12*a*b^5*c^2 +
48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^2 + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)
*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*(71680
*c^8*e^3*x^7 + 114688*a^2*c^6*d^3 + 5120*(48*c^8*d*e^2 + 25*b*c^7*e^3)*x^6 + 1280*(224*c^8*d^2*e + 352*b*c^7*d
*e^2 + (41*b^2*c^6 + 84*a*c^7)*e^3)*x^5 + 128*(896*c^8*d^3 + 4256*b*c^7*d^2*e + 32*(47*b^2*c^6 + 96*a*c^7)*d*e
^2 - 3*(b^3*c^5 - 188*a*b*c^6)*e^3)*x^4 - 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2*e + 32*(105*b^6
*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d*e^2 - (945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*
c^3 - 42432*a^3*b*c^4)*e^3 + 16*(14336*b*c^7*d^3 + 224*(69*b^2*c^6 + 140*a*c^7)*d^2*e - 32*(3*b^3*c^5 - 556*a*
b*c^6)*d*e^2 + (27*b^4*c^4 - 216*a*b^2*c^5 + 560*a^2*c^6)*e^3)*x^3 + 8*(14336*(b^2*c^6 + 2*a*c^7)*d^3 - 224*(b
^3*c^5 - 228*a*b*c^6)*d^2*e + 32*(7*b^4*c^4 - 60*a*b^2*c^5 + 192*a^2*c^6)*d*e^2 - (63*b^5*c^3 - 568*a*b^3*c^4
+ 1392*a^2*b*c^5)*e^3)*x^2 + 2*(114688*a*b*c^6*d^3 + 224*(5*b^4*c^4 - 48*a*b^2*c^5 + 240*a^2*c^6)*d^2*e - 32*(
35*b^5*c^3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d*e^2 + (315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^
3*c^5)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, -1/573440*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^
3*c^5)*d^2*e - 32*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^2 + (9*b^8 - 112*a*b^6*c + 480*a^
2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*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*(71680*c^8*e^3*x^7 + 114688*a^2*c^6*d^3 + 5120*(48*c^8*d*e^2 + 25*b*c^7*e^3)*x^6
+ 1280*(224*c^8*d^2*e + 352*b*c^7*d*e^2 + (41*b^2*c^6 + 84*a*c^7)*e^3)*x^5 + 128*(896*c^8*d^3 + 4256*b*c^7*d^
2*e + 32*(47*b^2*c^6 + 96*a*c^7)*d*e^2 - 3*(b^3*c^5 - 188*a*b*c^6)*e^3)*x^4 - 224*(15*b^5*c^3 - 160*a*b^3*c^4
+ 528*a^2*b*c^5)*d^2*e + 32*(105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d*e^2 - (945*b^7*
c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4)*e^3 + 16*(14336*b*c^7*d^3 + 224*(69*b^2*c^6 + 140*a
*c^7)*d^2*e - 32*(3*b^3*c^5 - 556*a*b*c^6)*d*e^2 + (27*b^4*c^4 - 216*a*b^2*c^5 + 560*a^2*c^6)*e^3)*x^3 + 8*(14
336*(b^2*c^6 + 2*a*c^7)*d^3 - 224*(b^3*c^5 - 228*a*b*c^6)*d^2*e + 32*(7*b^4*c^4 - 60*a*b^2*c^5 + 192*a^2*c^6)*
d*e^2 - (63*b^5*c^3 - 568*a*b^3*c^4 + 1392*a^2*b*c^5)*e^3)*x^2 + 2*(114688*a*b*c^6*d^3 + 224*(5*b^4*c^4 - 48*a
*b^2*c^5 + 240*a^2*c^6)*d^2*e - 32*(35*b^5*c^3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d*e^2 + (315*b^6*c^2 - 3164*a*
b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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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{3}{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)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.20642, size = 1156, normalized size = 3.05 \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)^(3/2),x, algorithm="giac")

[Out]

1/286720*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*c^2*x*e^3 + (48*c^9*d*e^2 + 25*b*c^8*e^3)/c^7)*x + (224*
c^9*d^2*e + 352*b*c^8*d*e^2 + 41*b^2*c^7*e^3 + 84*a*c^8*e^3)/c^7)*x + (896*c^9*d^3 + 4256*b*c^8*d^2*e + 1504*b
^2*c^7*d*e^2 + 3072*a*c^8*d*e^2 - 3*b^3*c^6*e^3 + 564*a*b*c^7*e^3)/c^7)*x + (14336*b*c^8*d^3 + 15456*b^2*c^7*d
^2*e + 31360*a*c^8*d^2*e - 96*b^3*c^6*d*e^2 + 17792*a*b*c^7*d*e^2 + 27*b^4*c^5*e^3 - 216*a*b^2*c^6*e^3 + 560*a
^2*c^7*e^3)/c^7)*x + (14336*b^2*c^7*d^3 + 28672*a*c^8*d^3 - 224*b^3*c^6*d^2*e + 51072*a*b*c^7*d^2*e + 224*b^4*
c^5*d*e^2 - 1920*a*b^2*c^6*d*e^2 + 6144*a^2*c^7*d*e^2 - 63*b^5*c^4*e^3 + 568*a*b^3*c^5*e^3 - 1392*a^2*b*c^6*e^
3)/c^7)*x + (114688*a*b*c^7*d^3 + 1120*b^4*c^5*d^2*e - 10752*a*b^2*c^6*d^2*e + 53760*a^2*c^7*d^2*e - 1120*b^5*
c^4*d*e^2 + 10752*a*b^3*c^5*d*e^2 - 29184*a^2*b*c^6*d*e^2 + 315*b^6*c^3*e^3 - 3164*a*b^4*c^4*e^3 + 9552*a^2*b^
2*c^5*e^3 - 6720*a^3*c^6*e^3)/c^7)*x + (114688*a^2*c^7*d^3 - 3360*b^5*c^4*d^2*e + 35840*a*b^3*c^5*d^2*e - 1182
72*a^2*b*c^6*d^2*e + 3360*b^6*c^3*d*e^2 - 35840*a*b^4*c^4*d*e^2 + 118272*a^2*b^2*c^5*d*e^2 - 98304*a^3*c^6*d*e
^2 - 945*b^7*c^2*e^3 + 10500*a*b^5*c^3*e^3 - 37744*a^2*b^3*c^4*e^3 + 42432*a^3*b*c^5*e^3)/c^7) - 3/16384*(32*b
^6*c^2*d^2*e - 384*a*b^4*c^3*d^2*e + 1536*a^2*b^2*c^4*d^2*e - 2048*a^3*c^5*d^2*e - 32*b^7*c*d*e^2 + 384*a*b^5*
c^2*d*e^2 - 1536*a^2*b^3*c^3*d*e^2 + 2048*a^3*b*c^4*d*e^2 + 9*b^8*e^3 - 112*a*b^6*c*e^3 + 480*a^2*b^4*c^2*e^3
- 768*a^3*b^2*c^3*e^3 + 256*a^4*c^4*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)