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

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

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Rubi [A]  time = 0.17, antiderivative size = 198, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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]

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*}

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Mathematica [A]  time = 0.52, size = 194, normalized size = 0.98 \begin {gather*} \frac {\frac {7}{24} e \left (b^2-4 a c\right ) \left (2 (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (\frac {3 \left (b^2-4 a c\right ) \left (\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 )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{128 c^{5/2}}+\frac {(b+2 c x) (a+x (b+c x))^{3/2}}{8 c}\right )\right )+2 c (a+x (b+c x))^{7/2} (2 c (8 d+7 e x)-b e)}{112 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*(a + x*(b + c*x))^(7/2)*(-(b*e) + 2*c*(8*d + 7*e*x)) + (7*(b^2 - 4*a*c)*e*(2*(b + 2*c*x)*(a + x*(b + c*x)
)^(5/2) - 5*(b^2 - 4*a*c)*(((b + 2*c*x)*(a + x*(b + c*x))^(3/2))/(8*c) + (3*(b^2 - 4*a*c)*(-2*Sqrt[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)])]))/(128*c^(5/2
)))))/24)/(112*c^2)

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IntegrateAlgebraic [B]  time = 1.97, size = 428, normalized size = 2.16 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-17856 a^3 b c^3 e+49152 a^3 c^4 d+13440 a^3 c^4 e x+8176 a^2 b^3 c^2 e-4512 a^2 b^2 c^3 e x+147456 a^2 b c^4 d x+84864 a^2 b c^4 e x^2+147456 a^2 c^5 d x^2+105728 a^2 c^5 e x^3-1540 a b^5 c e+952 a b^4 c^2 e x-704 a b^3 c^3 e x^2+147456 a b^2 c^4 d x^2+104320 a b^2 c^4 e x^3+294912 a b c^5 d x^3+230912 a b c^5 e x^4+147456 a c^6 d x^4+121856 a c^6 e x^5+105 b^7 e-70 b^6 c e x+56 b^5 c^2 e x^2-48 b^4 c^3 e x^3+49152 b^3 c^4 d x^3+38272 b^3 c^4 e x^4+147456 b^2 c^5 d x^4+121600 b^2 c^5 e x^5+147456 b c^6 d x^5+125952 b c^6 e x^6+49152 c^7 d x^6+43008 c^7 e x^7\right )}{172032 c^4}+\frac {5 e \left (256 a^4 c^4-256 a^3 b^2 c^3+96 a^2 b^4 c^2-16 a b^6 c+b^8\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{16384 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(49152*a^3*c^4*d + 105*b^7*e - 1540*a*b^5*c*e + 8176*a^2*b^3*c^2*e - 17856*a^3*b*c^3*e
+ 147456*a^2*b*c^4*d*x - 70*b^6*c*e*x + 952*a*b^4*c^2*e*x - 4512*a^2*b^2*c^3*e*x + 13440*a^3*c^4*e*x + 147456*
a*b^2*c^4*d*x^2 + 147456*a^2*c^5*d*x^2 + 56*b^5*c^2*e*x^2 - 704*a*b^3*c^3*e*x^2 + 84864*a^2*b*c^4*e*x^2 + 4915
2*b^3*c^4*d*x^3 + 294912*a*b*c^5*d*x^3 - 48*b^4*c^3*e*x^3 + 104320*a*b^2*c^4*e*x^3 + 105728*a^2*c^5*e*x^3 + 14
7456*b^2*c^5*d*x^4 + 147456*a*c^6*d*x^4 + 38272*b^3*c^4*e*x^4 + 230912*a*b*c^5*e*x^4 + 147456*b*c^6*d*x^5 + 12
1600*b^2*c^5*e*x^5 + 121856*a*c^6*e*x^5 + 49152*c^7*d*x^6 + 125952*b*c^6*e*x^6 + 43008*c^7*e*x^7))/(172032*c^4
) + (5*(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)*e*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2]])/(16384*c^(9/2))

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fricas [B]  time = 0.53, size = 841, normalized size = 4.25 \begin {gather*} \left [\frac {105 \, {\left (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}\right )} \sqrt {c} e \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (43008 \, c^{8} e x^{7} + 49152 \, a^{3} c^{5} d + 3072 \, {\left (16 \, c^{8} d + 41 \, b c^{7} e\right )} x^{6} + 256 \, {\left (576 \, b c^{7} d + {\left (475 \, b^{2} c^{6} + 476 \, a c^{7}\right )} e\right )} x^{5} + 128 \, {\left (1152 \, {\left (b^{2} c^{6} + a c^{7}\right )} d + {\left (299 \, b^{3} c^{5} + 1804 \, a b c^{6}\right )} e\right )} x^{4} + 16 \, {\left (3072 \, {\left (b^{3} c^{5} + 6 \, a b c^{6}\right )} d - {\left (3 \, b^{4} c^{4} - 6520 \, a b^{2} c^{5} - 6608 \, a^{2} c^{6}\right )} e\right )} x^{3} + 8 \, {\left (18432 \, {\left (a b^{2} c^{5} + a^{2} c^{6}\right )} d + {\left (7 \, b^{5} c^{3} - 88 \, a b^{3} c^{4} + 10608 \, a^{2} b c^{5}\right )} e\right )} x^{2} + {\left (105 \, b^{7} c - 1540 \, a b^{5} c^{2} + 8176 \, a^{2} b^{3} c^{3} - 17856 \, a^{3} b c^{4}\right )} e + 2 \, {\left (73728 \, a^{2} b c^{5} d - {\left (35 \, b^{6} c^{2} - 476 \, a b^{4} c^{3} + 2256 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{688128 \, c^{5}}, \frac {105 \, {\left (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}\right )} \sqrt {-c} e \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (43008 \, c^{8} e x^{7} + 49152 \, a^{3} c^{5} d + 3072 \, {\left (16 \, c^{8} d + 41 \, b c^{7} e\right )} x^{6} + 256 \, {\left (576 \, b c^{7} d + {\left (475 \, b^{2} c^{6} + 476 \, a c^{7}\right )} e\right )} x^{5} + 128 \, {\left (1152 \, {\left (b^{2} c^{6} + a c^{7}\right )} d + {\left (299 \, b^{3} c^{5} + 1804 \, a b c^{6}\right )} e\right )} x^{4} + 16 \, {\left (3072 \, {\left (b^{3} c^{5} + 6 \, a b c^{6}\right )} d - {\left (3 \, b^{4} c^{4} - 6520 \, a b^{2} c^{5} - 6608 \, a^{2} c^{6}\right )} e\right )} x^{3} + 8 \, {\left (18432 \, {\left (a b^{2} c^{5} + a^{2} c^{6}\right )} d + {\left (7 \, b^{5} c^{3} - 88 \, a b^{3} c^{4} + 10608 \, a^{2} b c^{5}\right )} e\right )} x^{2} + {\left (105 \, b^{7} c - 1540 \, a b^{5} c^{2} + 8176 \, a^{2} b^{3} c^{3} - 17856 \, a^{3} b c^{4}\right )} e + 2 \, {\left (73728 \, a^{2} b c^{5} d - {\left (35 \, b^{6} c^{2} - 476 \, a b^{4} c^{3} + 2256 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{344064 \, c^{5}}\right ] \end {gather*}

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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giac [B]  time = 0.27, size = 452, normalized size = 2.28 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.06, size = 616, normalized size = 3.11 \begin {gather*} -\frac {5 a^{4} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 \sqrt {c}}+\frac {5 a^{3} b^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {3}{2}}}-\frac {15 a^{2} b^{4} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {5}{2}}}+\frac {5 a \,b^{6} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {7}{2}}}-\frac {5 b^{8} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16384 c^{\frac {9}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{3} e x}{64}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} e x}{256 c}-\frac {15 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} e x}{1024 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{6} e x}{4096 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{3} b e}{128 c}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} e}{512 c^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} e x}{96}-\frac {15 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} e}{2048 c^{3}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} e x}{192 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{7} e}{8192 c^{4}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} e x}{1536 c^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b e}{192 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{3} e}{384 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a e x}{24}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{5} e}{3072 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} e x}{96 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b e}{48 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{3} e}{192 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} e x}{4}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} b e}{56 c}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} d}{7} \end {gather*}

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]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (b+2\,c\,x\right )\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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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