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

Optimal. Leaf size=118 \[ \frac {16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {722, 636} \begin {gather*} \frac {16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (16*(c*d^2 - b*d*
e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 722

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (8 \left (c d^2-b d e+a e^2\right )\right ) \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {16 \left (c d^2-b d e+a e^2\right ) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.73, size = 190, normalized size = 1.61 \begin {gather*} \frac {2 \left (12 b (d-e x) \left (2 a^2 e^2+a c (d-e x)^2+2 c^2 d^2 x^2\right )-8 \left (2 a^3 e^3+3 a^2 c e \left (d^2+e^2 x^2\right )-3 a c^2 d x \left (d^2+e^2 x^2\right )-2 c^3 d^3 x^3\right )-6 b^2 \left (d^2-6 d e x+e^2 x^2\right ) (a e-c d x)+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + 12*b*(d
- e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^2 + a*c*(d - e*x)^2) - 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e^2*x^2
) - 3*a*c^2*d*x*(d^2 + e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))

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IntegrateAlgebraic [B]  time = 1.73, size = 288, normalized size = 2.44 \begin {gather*} -\frac {2 \left (16 a^3 e^3-24 a^2 b d e^2+24 a^2 b e^3 x+24 a^2 c d^2 e+24 a^2 c e^3 x^2+6 a b^2 d^2 e-36 a b^2 d e^2 x+6 a b^2 e^3 x^2-12 a b c d^3+36 a b c d^2 e x-36 a b c d e^2 x^2+12 a b c e^3 x^3-24 a c^2 d^3 x-24 a c^2 d e^2 x^3+b^3 d^3+9 b^3 d^2 e x-9 b^3 d e^2 x^2-b^3 e^3 x^3-6 b^2 c d^3 x+36 b^2 c d^2 e x^2-6 b^2 c d e^2 x^3-24 b c^2 d^3 x^2+24 b c^2 d^2 e x^3-16 c^3 d^3 x^3\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^3*d^3 - 12*a*b*c*d^3 + 6*a*b^2*d^2*e + 24*a^2*c*d^2*e - 24*a^2*b*d*e^2 + 16*a^3*e^3 - 6*b^2*c*d^3*x - 2
4*a*c^2*d^3*x + 9*b^3*d^2*e*x + 36*a*b*c*d^2*e*x - 36*a*b^2*d*e^2*x + 24*a^2*b*e^3*x - 24*b*c^2*d^3*x^2 + 36*b
^2*c*d^2*e*x^2 - 9*b^3*d*e^2*x^2 - 36*a*b*c*d*e^2*x^2 + 6*a*b^2*e^3*x^2 + 24*a^2*c*e^3*x^2 - 16*c^3*d^3*x^3 +
24*b*c^2*d^2*e*x^3 - 6*b^2*c*d*e^2*x^3 - 24*a*c^2*d*e^2*x^3 - b^3*e^3*x^3 + 12*a*b*c*e^3*x^3))/(3*(b^2 - 4*a*c
)^2*(a + b*x + c*x^2)^(3/2))

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fricas [B]  time = 1.19, size = 366, normalized size = 3.10 \begin {gather*} \frac {2 \, {\left (24 \, a^{2} b d e^{2} - 16 \, a^{3} e^{3} - {\left (b^{3} - 12 \, a b c\right )} d^{3} - 6 \, {\left (a b^{2} + 4 \, a^{2} c\right )} d^{2} e + {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} x^{3} + 3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, {\left (b^{3} + 4 \, a b c\right )} d e^{2} - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e^{3}\right )} x^{2} + 3 \, {\left (12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3} + 2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{3} - 3 \, {\left (b^{3} + 4 \, a b c\right )} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(24*a^2*b*d*e^2 - 16*a^3*e^3 - (b^3 - 12*a*b*c)*d^3 - 6*(a*b^2 + 4*a^2*c)*d^2*e + (16*c^3*d^3 - 24*b*c^2*d
^2*e + 6*(b^2*c + 4*a*c^2)*d*e^2 + (b^3 - 12*a*b*c)*e^3)*x^3 + 3*(8*b*c^2*d^3 - 12*b^2*c*d^2*e + 3*(b^3 + 4*a*
b*c)*d*e^2 - 2*(a*b^2 + 4*a^2*c)*e^3)*x^2 + 3*(12*a*b^2*d*e^2 - 8*a^2*b*e^3 + 2*(b^2*c + 4*a*c^2)*d^3 - 3*(b^3
+ 4*a*b*c)*d^2*e)*x)*sqrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*
a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*
a^2*b^3*c + 16*a^3*b*c^2)*x)

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giac [B]  time = 0.31, size = 327, normalized size = 2.77 \begin {gather*} \frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + 24 \, a c^{2} d e^{2} + b^{3} e^{3} - 12 \, a b c e^{3}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2} + 12 \, a b c d e^{2} - 2 \, a b^{2} e^{3} - 8 \, a^{2} c e^{3}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d^{3} + 8 \, a c^{2} d^{3} - 3 \, b^{3} d^{2} e - 12 \, a b c d^{2} e + 12 \, a b^{2} d e^{2} - 8 \, a^{2} b e^{3}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d^{3} - 12 \, a b c d^{3} + 6 \, a b^{2} d^{2} e + 24 \, a^{2} c d^{2} e - 24 \, a^{2} b d e^{2} + 16 \, a^{3} e^{3}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((((16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + 24*a*c^2*d*e^2 + b^3*e^3 - 12*a*b*c*e^3)*x/(b^4 - 8*a*b^
2*c + 16*a^2*c^2) + 3*(8*b*c^2*d^3 - 12*b^2*c*d^2*e + 3*b^3*d*e^2 + 12*a*b*c*d*e^2 - 2*a*b^2*e^3 - 8*a^2*c*e^3
)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 3*(2*b^2*c*d^3 + 8*a*c^2*d^3 - 3*b^3*d^2*e - 12*a*b*c*d^2*e + 12*a*b^2*d
*e^2 - 8*a^2*b*e^3)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x - (b^3*d^3 - 12*a*b*c*d^3 + 6*a*b^2*d^2*e + 24*a^2*c*d^2
*e - 24*a^2*b*d*e^2 + 16*a^3*e^3)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a)^(3/2)

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maple [B]  time = 0.11, size = 296, normalized size = 2.51 \begin {gather*} -\frac {2 \left (12 a b c \,e^{3} x^{3}-24 a \,c^{2} d \,e^{2} x^{3}-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 c^{3} d^{3} x^{3}+24 a^{2} c \,e^{3} x^{2}+6 a \,b^{2} e^{3} x^{2}-36 a b c d \,e^{2} x^{2}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+24 a^{2} b \,e^{3} x -36 a \,b^{2} d \,e^{2} x +36 a b c \,d^{2} e x -24 a \,c^{2} d^{3} x +9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +16 a^{3} e^{3}-24 a^{2} b d \,e^{2}+24 a^{2} c \,d^{2} e +6 a \,b^{2} d^{2} e -12 a b c \,d^{3}+b^{3} d^{3}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(c*x^2+b*x+a)^(3/2)*(12*a*b*c*e^3*x^3-24*a*c^2*d*e^2*x^3-b^3*e^3*x^3-6*b^2*c*d*e^2*x^3+24*b*c^2*d^2*e*x^3
-16*c^3*d^3*x^3+24*a^2*c*e^3*x^2+6*a*b^2*e^3*x^2-36*a*b*c*d*e^2*x^2-9*b^3*d*e^2*x^2+36*b^2*c*d^2*e*x^2-24*b*c^
2*d^3*x^2+24*a^2*b*e^3*x-36*a*b^2*d*e^2*x+36*a*b*c*d^2*e*x-24*a*c^2*d^3*x+9*b^3*d^2*e*x-6*b^2*c*d^3*x+16*a^3*e
^3-24*a^2*b*d*e^2+24*a^2*c*d^2*e+6*a*b^2*d^2*e-12*a*b*c*d^3+b^3*d^3)/(16*a^2*c^2-8*a*b^2*c+b^4)

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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((e*x+d)^3/(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 zero or nonzero?

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mupad [B]  time = 1.62, size = 528, normalized size = 4.47 \begin {gather*} \frac {2\,a\,b^4\,e^3+2\,b^5\,e^3\,x-2\,b^4\,e^3\,\left (c\,x^2+b\,x+a\right )+16\,a^3\,c^2\,e^3-2\,b^3\,c^2\,d^3-12\,a^2\,b^2\,c\,e^3-48\,a^2\,c^3\,d^2\,e-4\,b^2\,c^3\,d^3\,x-48\,a^2\,c^2\,e^3\,\left (c\,x^2+b\,x+a\right )+8\,a\,b\,c^3\,d^3+16\,a\,c^4\,d^3\,x+16\,b\,c^3\,d^3\,\left (c\,x^2+b\,x+a\right )+32\,c^4\,d^3\,x\,\left (c\,x^2+b\,x+a\right )-6\,a\,b^3\,c\,d\,e^2-14\,a\,b^3\,c\,e^3\,x-6\,b^4\,c\,d\,e^2\,x+12\,a\,b^2\,c\,e^3\,\left (c\,x^2+b\,x+a\right )+6\,b^3\,c\,d\,e^2\,\left (c\,x^2+b\,x+a\right )+2\,b^3\,c\,e^3\,x\,\left (c\,x^2+b\,x+a\right )+12\,a\,b^2\,c^2\,d^2\,e+24\,a^2\,b\,c^2\,d\,e^2+24\,a^2\,b\,c^2\,e^3\,x-48\,a^2\,c^3\,d\,e^2\,x+6\,b^3\,c^2\,d^2\,e\,x-24\,b^2\,c^2\,d^2\,e\,\left (c\,x^2+b\,x+a\right )+12\,b^2\,c^2\,d\,e^2\,x\,\left (c\,x^2+b\,x+a\right )-24\,a\,b\,c^3\,d^2\,e\,x+24\,a\,b\,c^2\,d\,e^2\,\left (c\,x^2+b\,x+a\right )-24\,a\,b\,c^2\,e^3\,x\,\left (c\,x^2+b\,x+a\right )+48\,a\,c^3\,d\,e^2\,x\,\left (c\,x^2+b\,x+a\right )-48\,b\,c^3\,d^2\,e\,x\,\left (c\,x^2+b\,x+a\right )+36\,a\,b^2\,c^2\,d\,e^2\,x}{\left (48\,a^2\,c^4-24\,a\,b^2\,c^3+3\,b^4\,c^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*b^4*e^3 + 2*b^5*e^3*x - 2*b^4*e^3*(a + b*x + c*x^2) + 16*a^3*c^2*e^3 - 2*b^3*c^2*d^3 - 12*a^2*b^2*c*e^3 -
48*a^2*c^3*d^2*e - 4*b^2*c^3*d^3*x - 48*a^2*c^2*e^3*(a + b*x + c*x^2) + 8*a*b*c^3*d^3 + 16*a*c^4*d^3*x + 16*b
*c^3*d^3*(a + b*x + c*x^2) + 32*c^4*d^3*x*(a + b*x + c*x^2) - 6*a*b^3*c*d*e^2 - 14*a*b^3*c*e^3*x - 6*b^4*c*d*e
^2*x + 12*a*b^2*c*e^3*(a + b*x + c*x^2) + 6*b^3*c*d*e^2*(a + b*x + c*x^2) + 2*b^3*c*e^3*x*(a + b*x + c*x^2) +
12*a*b^2*c^2*d^2*e + 24*a^2*b*c^2*d*e^2 + 24*a^2*b*c^2*e^3*x - 48*a^2*c^3*d*e^2*x + 6*b^3*c^2*d^2*e*x - 24*b^2
*c^2*d^2*e*(a + b*x + c*x^2) + 12*b^2*c^2*d*e^2*x*(a + b*x + c*x^2) - 24*a*b*c^3*d^2*e*x + 24*a*b*c^2*d*e^2*(a
+ b*x + c*x^2) - 24*a*b*c^2*e^3*x*(a + b*x + c*x^2) + 48*a*c^3*d*e^2*x*(a + b*x + c*x^2) - 48*b*c^3*d^2*e*x*(
a + b*x + c*x^2) + 36*a*b^2*c^2*d*e^2*x)/((48*a^2*c^4 + 3*b^4*c^2 - 24*a*b^2*c^3)*(a + b*x + c*x^2)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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