3.24.94 \(\int \frac {(d+e x)^2}{(a+b x+c x^2)^{5/2}} \, dx\) [2394]

Optimal. Leaf size=98 \[ -\frac {2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]

[Out]

-2/3*(2*c*x+b)*(e*x+d)^2/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+8/3*(-b*e+2*c*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+
b^2)^2/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 98, 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 = {742, 650} \begin {gather*} \frac {8 (2 c d-b e) (-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 (b+2 c x) (d+e x)^2}{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)^2/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b + 2*c*x)*(d + e*x)^2)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (8*(2*c*d - b*e)*(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 650

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 742

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {(4 (2 c d-b e)) \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 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (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 = 0.89, size = 167, normalized size = 1.70 \begin {gather*} \frac {2 \left (-b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b \left (2 a^2 e^2+2 c^2 d x^2 (3 d-2 e x)+3 a c (d-e x)^2\right )+8 c \left (-2 a^2 d e+2 c^2 d^2 x^3+a c x \left (3 d^2+e^2 x^2\right )\right )+b^2 \left (-4 a e (d-3 e x)+2 c x \left (3 d^2-12 d e x+e^2 x^2\right )\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)^2/(a + b*x + c*x^2)^(5/2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(90)=180\).
time = 0.80, size = 357, normalized size = 3.64

method	result	size
trager	\(\frac {\frac {16}{3} a \,c^{2} e^{2} x^{3}+\frac {4}{3} b^{2} c \,e^{2} x^{3}-\frac {32}{3} b \,c^{2} d e \,x^{3}+\frac {32}{3} c^{3} d^{2} x^{3}+8 a b c \,e^{2} x^{2}+2 b^{3} e^{2} x^{2}-16 b^{2} c d e \,x^{2}+16 b \,c^{2} d^{2} x^{2}+8 a \,b^{2} e^{2} x -16 a b c d e x +16 a \,c^{2} d^{2} x -4 b^{3} d e x +4 b^{2} c \,d^{2} x +\frac {16}{3} a^{2} e^{2} b -\frac {32}{3} a^{2} c d e -\frac {8}{3} a \,b^{2} d e +8 a b c \,d^{2}-\frac {2}{3} b^{3} d^{2}}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(206\)
gosper	\(\frac {\frac {16}{3} a \,c^{2} e^{2} x^{3}+\frac {4}{3} b^{2} c \,e^{2} x^{3}-\frac {32}{3} b \,c^{2} d e \,x^{3}+\frac {32}{3} c^{3} d^{2} x^{3}+8 a b c \,e^{2} x^{2}+2 b^{3} e^{2} x^{2}-16 b^{2} c d e \,x^{2}+16 b \,c^{2} d^{2} x^{2}+8 a \,b^{2} e^{2} x -16 a b c d e x +16 a \,c^{2} d^{2} x -4 b^{3} d e x +4 b^{2} c \,d^{2} x +\frac {16}{3} a^{2} e^{2} b -\frac {32}{3} a^{2} c d e -\frac {8}{3} a \,b^{2} d e +8 a b c \,d^{2}-\frac {2}{3} b^{3} d^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}\)	\(215\)
default	\(e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+2 d e \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+d^{2} \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )\)	\(357\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^
2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+
b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*d*e*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2
/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+d^2*(2/3*(2*
c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(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((e*x+d)^2/(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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (94) = 188\).
time = 5.01, size = 306, normalized size = 3.12 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{2} x^{3} + 24 \, b c^{2} d^{2} x^{2} + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} x - {\left (b^{3} - 12 \, a b c\right )} d^{2} + {\left (12 \, a b^{2} x + 2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} x^{3} + 8 \, a^{2} b + 3 \, {\left (b^{3} + 4 \, a b c\right )} x^{2}\right )} e^{2} - 2 \, {\left (8 \, b c^{2} d x^{3} + 12 \, b^{2} c d x^{2} + 3 \, {\left (b^{3} + 4 \, a b c\right )} d x + 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} d\right )} e\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)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(16*c^3*d^2*x^3 + 24*b*c^2*d^2*x^2 + 6*(b^2*c + 4*a*c^2)*d^2*x - (b^3 - 12*a*b*c)*d^2 + (12*a*b^2*x + 2*(b
^2*c + 4*a*c^2)*x^3 + 8*a^2*b + 3*(b^3 + 4*a*b*c)*x^2)*e^2 - 2*(8*b*c^2*d*x^3 + 12*b^2*c*d*x^2 + 3*(b^3 + 4*a*
b*c)*d*x + 2*(a*b^2 + 4*a^2*c)*d)*e)*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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Sympy [F(-1)] Timed out
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)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (94) = 188\).
time = 2.29, size = 263, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2} + 4 \, a c^{2} e^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2} + 4 \, a b c e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {6 \, {\left (b^{2} c d^{2} + 4 \, a c^{2} d^{2} - b^{3} d e - 4 \, a b c d e + 2 \, a b^{2} e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d^{2} - 12 \, a b c d^{2} + 4 \, a b^{2} d e + 16 \, a^{2} c d e - 8 \, a^{2} b e^{2}}{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)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.42, size = 321, normalized size = 3.28 \begin {gather*} \frac {2\,b^3\,e^2\,\left (c\,x^2+b\,x+a\right )-2\,b^3\,c\,d^2-2\,b^4\,e^2\,x-2\,a\,b^3\,e^2-16\,a^2\,c^2\,e^2\,x-4\,b^2\,c^2\,d^2\,x+8\,a\,b\,c^2\,d^2+8\,a^2\,b\,c\,e^2-32\,a^2\,c^2\,d\,e+16\,a\,c^3\,d^2\,x+16\,b\,c^2\,d^2\,\left (c\,x^2+b\,x+a\right )+32\,c^3\,d^2\,x\,\left (c\,x^2+b\,x+a\right )+12\,a\,b^2\,c\,e^2\,x+16\,a\,c^2\,e^2\,x\,\left (c\,x^2+b\,x+a\right )+4\,b^2\,c\,e^2\,x\,\left (c\,x^2+b\,x+a\right )+8\,a\,b^2\,c\,d\,e+4\,b^3\,c\,d\,e\,x+8\,a\,b\,c\,e^2\,\left (c\,x^2+b\,x+a\right )-16\,b^2\,c\,d\,e\,\left (c\,x^2+b\,x+a\right )-16\,a\,b\,c^2\,d\,e\,x-32\,b\,c^2\,d\,e\,x\,\left (c\,x^2+b\,x+a\right )}{\left (48\,a^2\,c^3-24\,a\,b^2\,c^2+3\,b^4\,c\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)^2/(a + b*x + c*x^2)^(5/2),x)

[Out]

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

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