3.1567 \(\int (b+2 c x) (d+e x) (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=198 \[ \frac{5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^4}-\frac{5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^3}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}-\frac{5 e \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{9/2}}+\frac{\left (a+b x+c x^2\right )^{7/2} (-b e+16 c d+14 c e x)}{56 c} \]

[Out]

(5*(b^2 - 4*a*c)^3*e*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^4) - (5*(b^2 - 4*a*c)^2*e*(b + 2*c*x)*(a + b*x
+ c*x^2)^(3/2))/(3072*c^3) + ((b^2 - 4*a*c)*e*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(192*c^2) + ((16*c*d - b*e
+ 14*c*e*x)*(a + b*x + c*x^2)^(7/2))/(56*c) - (5*(b^2 - 4*a*c)^4*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*
x + c*x^2])])/(16384*c^(9/2))

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Rubi [A]  time = 0.16917, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {779, 612, 621, 206} \[ \frac{5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^4}-\frac{5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^3}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}-\frac{5 e \left (b^2-4 a c\right )^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{9/2}}+\frac{\left (a+b x+c x^2\right )^{7/2} (-b e+16 c d+14 c e x)}{56 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^2 - 4*a*c)^3*e*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^4) - (5*(b^2 - 4*a*c)^2*e*(b + 2*c*x)*(a + b*x
+ c*x^2)^(3/2))/(3072*c^3) + ((b^2 - 4*a*c)*e*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(192*c^2) + ((16*c*d - b*e
+ 14*c*e*x)*(a + b*x + c*x^2)^(7/2))/(56*c) - (5*(b^2 - 4*a*c)^4*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*
x + c*x^2])])/(16384*c^(9/2))

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) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{(16 c d-b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{56 c}+\frac{\left (\left (b^2-4 a c\right ) e\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c}\\ &=\frac{\left (b^2-4 a c\right ) e (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}+\frac{(16 c d-b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{56 c}-\frac{\left (5 \left (b^2-4 a c\right )^2 e\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{384 c^2}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 e (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^3}+\frac{\left (b^2-4 a c\right ) e (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}+\frac{(16 c d-b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{56 c}+\frac{\left (5 \left (b^2-4 a c\right )^3 e\right ) \int \sqrt{a+b x+c x^2} \, dx}{2048 c^3}\\ &=\frac{5 \left (b^2-4 a c\right )^3 e (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^4}-\frac{5 \left (b^2-4 a c\right )^2 e (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^3}+\frac{\left (b^2-4 a c\right ) e (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}+\frac{(16 c d-b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{56 c}-\frac{\left (5 \left (b^2-4 a c\right )^4 e\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16384 c^4}\\ &=\frac{5 \left (b^2-4 a c\right )^3 e (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^4}-\frac{5 \left (b^2-4 a c\right )^2 e (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^3}+\frac{\left (b^2-4 a c\right ) e (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}+\frac{(16 c d-b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{56 c}-\frac{\left (5 \left (b^2-4 a c\right )^4 e\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^4}\\ &=\frac{5 \left (b^2-4 a c\right )^3 e (b+2 c x) \sqrt{a+b x+c x^2}}{8192 c^4}-\frac{5 \left (b^2-4 a c\right )^2 e (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^3}+\frac{\left (b^2-4 a c\right ) e (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^2}+\frac{(16 c d-b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{56 c}-\frac{5 \left (b^2-4 a c\right )^4 e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.316063, size = 195, normalized size = 0.98 \[ \frac{e \left (b^2-4 a c\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{49152 c^{9/2}}+\frac{(a+x (b+c x))^{7/2} (2 c (8 d+7 e x)-b e)}{56 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + x*(b + c*x))^(7/2)*(-(b*e) + 2*c*(8*d + 7*e*x)))/(56*c) + ((b^2 - 4*a*c)*e*(256*c^(5/2)*(b + 2*c*x)*(a +
x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqr
t[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])
)))/(49152*c^(9/2))

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Maple [B]  time = 0.009, size = 616, normalized size = 3.1 \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)*(c*x^2+b*x+a)^(5/2),x)

[Out]

2/7*(c*x^2+b*x+a)^(7/2)*d-5/1536/c^2*e*b^4*(c*x^2+b*x+a)^(3/2)*x-1/48/c*e*a*(c*x^2+b*x+a)^(5/2)*b+5/192/c*e*b^
2*(c*x^2+b*x+a)^(3/2)*x*a+15/256/c*e*b^2*(c*x^2+b*x+a)^(1/2)*x*a^2-15/1024/c^2*e*b^4*(c*x^2+b*x+a)^(1/2)*x*a-1
/24*e*a*(c*x^2+b*x+a)^(5/2)*x-5/3072/c^3*e*b^5*(c*x^2+b*x+a)^(3/2)-5/64/c^(1/2)*e*a^4*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))-5/16384/c^(9/2)*e*b^8*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/8192/c^4*e*b^7*(c*x^2+
b*x+a)^(1/2)-1/56/c*e*b*(c*x^2+b*x+a)^(7/2)+1/192/c^2*e*b^3*(c*x^2+b*x+a)^(5/2)-5/64*e*a^3*(c*x^2+b*x+a)^(1/2)
*x-5/96*e*a^2*(c*x^2+b*x+a)^(3/2)*x+1/4*e*x*(c*x^2+b*x+a)^(7/2)+15/512/c^2*e*b^3*(c*x^2+b*x+a)^(1/2)*a^2+5/64/
c^(3/2)*e*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+5/1024/c^(7/2)*e*b^6*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a-15/512/c^(5/2)*e*b^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-5/192/c*e*a^2*(c*x^2+b
*x+a)^(3/2)*b+5/384/c^2*e*b^3*(c*x^2+b*x+a)^(3/2)*a+5/4096/c^3*e*b^6*(c*x^2+b*x+a)^(1/2)*x+1/96/c*e*b^2*(c*x^2
+b*x+a)^(5/2)*x-15/2048/c^3*e*b^5*(c*x^2+b*x+a)^(1/2)*a-5/128/c*e*a^3*(c*x^2+b*x+a)^(1/2)*b

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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)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.3279, size = 2009, normalized size = 10.15 \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)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/688128*(105*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(c)*e*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*(43008*c^8*e*x^7 + 49152*a^3*c^5*d +
3072*(16*c^8*d + 41*b*c^7*e)*x^6 + 256*(576*b*c^7*d + (475*b^2*c^6 + 476*a*c^7)*e)*x^5 + 128*(1152*(b^2*c^6 +
a*c^7)*d + (299*b^3*c^5 + 1804*a*b*c^6)*e)*x^4 + 16*(3072*(b^3*c^5 + 6*a*b*c^6)*d - (3*b^4*c^4 - 6520*a*b^2*c^
5 - 6608*a^2*c^6)*e)*x^3 + 8*(18432*(a*b^2*c^5 + a^2*c^6)*d + (7*b^5*c^3 - 88*a*b^3*c^4 + 10608*a^2*b*c^5)*e)*
x^2 + (105*b^7*c - 1540*a*b^5*c^2 + 8176*a^2*b^3*c^3 - 17856*a^3*b*c^4)*e + 2*(73728*a^2*b*c^5*d - (35*b^6*c^2
- 476*a*b^4*c^3 + 2256*a^2*b^2*c^4 - 6720*a^3*c^5)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/344064*(105*(b^8 - 16*
a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-c)*e*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*(43008*c^8*e*x^7 + 49152*a^3*c^5*d + 3072*(16*c^8*d + 41*b*c^7*e)*x^
6 + 256*(576*b*c^7*d + (475*b^2*c^6 + 476*a*c^7)*e)*x^5 + 128*(1152*(b^2*c^6 + a*c^7)*d + (299*b^3*c^5 + 1804*
a*b*c^6)*e)*x^4 + 16*(3072*(b^3*c^5 + 6*a*b*c^6)*d - (3*b^4*c^4 - 6520*a*b^2*c^5 - 6608*a^2*c^6)*e)*x^3 + 8*(1
8432*(a*b^2*c^5 + a^2*c^6)*d + (7*b^5*c^3 - 88*a*b^3*c^4 + 10608*a^2*b*c^5)*e)*x^2 + (105*b^7*c - 1540*a*b^5*c
^2 + 8176*a^2*b^3*c^3 - 17856*a^3*b*c^4)*e + 2*(73728*a^2*b*c^5*d - (35*b^6*c^2 - 476*a*b^4*c^3 + 2256*a^2*b^2
*c^4 - 6720*a^3*c^5)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5]

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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 ) \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)*(c*x**2+b*x+a)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.34364, size = 610, normalized size = 3.08 \begin{align*} \frac{1}{172032} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, c^{3} x e + \frac{16 \, c^{10} d + 41 \, b c^{9} e}{c^{7}}\right )} x + \frac{576 \, b c^{9} d + 475 \, b^{2} c^{8} e + 476 \, a c^{9} e}{c^{7}}\right )} x + \frac{1152 \, b^{2} c^{8} d + 1152 \, a c^{9} d + 299 \, b^{3} c^{7} e + 1804 \, a b c^{8} e}{c^{7}}\right )} x + \frac{3072 \, b^{3} c^{7} d + 18432 \, a b c^{8} d - 3 \, b^{4} c^{6} e + 6520 \, a b^{2} c^{7} e + 6608 \, a^{2} c^{8} e}{c^{7}}\right )} x + \frac{18432 \, a b^{2} c^{7} d + 18432 \, a^{2} c^{8} d + 7 \, b^{5} c^{5} e - 88 \, a b^{3} c^{6} e + 10608 \, a^{2} b c^{7} e}{c^{7}}\right )} x + \frac{73728 \, a^{2} b c^{7} d - 35 \, b^{6} c^{4} e + 476 \, a b^{4} c^{5} e - 2256 \, a^{2} b^{2} c^{6} e + 6720 \, a^{3} c^{7} e}{c^{7}}\right )} x + \frac{49152 \, a^{3} c^{7} d + 105 \, b^{7} c^{3} e - 1540 \, a b^{5} c^{4} e + 8176 \, a^{2} b^{3} c^{5} e - 17856 \, a^{3} b c^{6} e}{c^{7}}\right )} + \frac{5 \,{\left (b^{8} e - 16 \, a b^{6} c e + 96 \, a^{2} b^{4} c^{2} e - 256 \, a^{3} b^{2} c^{3} e + 256 \, a^{4} c^{4} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16384 \, c^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/172032*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*c^3*x*e + (16*c^10*d + 41*b*c^9*e)/c^7)*x + (576*b*c^9*d
+ 475*b^2*c^8*e + 476*a*c^9*e)/c^7)*x + (1152*b^2*c^8*d + 1152*a*c^9*d + 299*b^3*c^7*e + 1804*a*b*c^8*e)/c^7)
*x + (3072*b^3*c^7*d + 18432*a*b*c^8*d - 3*b^4*c^6*e + 6520*a*b^2*c^7*e + 6608*a^2*c^8*e)/c^7)*x + (18432*a*b^
2*c^7*d + 18432*a^2*c^8*d + 7*b^5*c^5*e - 88*a*b^3*c^6*e + 10608*a^2*b*c^7*e)/c^7)*x + (73728*a^2*b*c^7*d - 35
*b^6*c^4*e + 476*a*b^4*c^5*e - 2256*a^2*b^2*c^6*e + 6720*a^3*c^7*e)/c^7)*x + (49152*a^3*c^7*d + 105*b^7*c^3*e
- 1540*a*b^5*c^4*e + 8176*a^2*b^3*c^5*e - 17856*a^3*b*c^6*e)/c^7) + 5/16384*(b^8*e - 16*a*b^6*c*e + 96*a^2*b^4
*c^2*e - 256*a^3*b^2*c^3*e + 256*a^4*c^4*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/
2)