3.17.28 \(\int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx\) [1628]

Optimal. Leaf size=69 \[ -\frac {2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}+\frac {4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2}{e^3 \sqrt {d+e x}} \]

[Out]

-2/5*(-a*e+b*d)^2/e^3/(e*x+d)^(5/2)+4/3*b*(-a*e+b*d)/e^3/(e*x+d)^(3/2)-2*b^2/e^3/(e*x+d)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \begin {gather*} \frac {4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 b^2}{e^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*b*(b*d - a*e))/(3*e^3*(d + e*x)^(3/2)) - (2*b^2)/(e^3*Sqrt[d +
e*x])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^{7/2}}-\frac {2 b (b d-a e)}{e^2 (d+e x)^{5/2}}+\frac {b^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}+\frac {4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2}{e^3 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 61, normalized size = 0.88 \begin {gather*} -\frac {2 \left (3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(3*a^2*e^2 + 2*a*b*e*(2*d + 5*e*x) + b^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))

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Maple [A]
time = 0.47, size = 67, normalized size = 0.97

method	result	size
gosper	\(-\frac {2 \left (15 b^{2} x^{2} e^{2}+10 a b \,e^{2} x +20 b^{2} d e x +3 a^{2} e^{2}+4 a b d e +8 b^{2} d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}}\)	\(63\)
trager	\(-\frac {2 \left (15 b^{2} x^{2} e^{2}+10 a b \,e^{2} x +20 b^{2} d e x +3 a^{2} e^{2}+4 a b d e +8 b^{2} d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}}\)	\(63\)
derivativedivides	\(\frac {-\frac {4 b \left (a e -b d \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b^{2}}{\sqrt {e x +d}}}{e^{3}}\)	\(67\)
default	\(\frac {-\frac {4 b \left (a e -b d \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b^{2}}{\sqrt {e x +d}}}{e^{3}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/e^3*(-2/3*b*(a*e-b*d)/(e*x+d)^(3/2)-1/5*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(e*x+d)^(5/2)-b^2/(e*x+d)^(1/2))

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Maxima [A]
time = 0.27, size = 68, normalized size = 0.99 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} b^{2} + 3 \, b^{2} d^{2} - 6 \, a b d e + 3 \, a^{2} e^{2} - 10 \, {\left (b^{2} d - a b e\right )} {\left (x e + d\right )}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

-2/15*(15*(x*e + d)^2*b^2 + 3*b^2*d^2 - 6*a*b*d*e + 3*a^2*e^2 - 10*(b^2*d - a*b*e)*(x*e + d))*e^(-3)/(x*e + d)
^(5/2)

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Fricas [A]
time = 2.78, size = 88, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{2} + {\left (15 \, b^{2} x^{2} + 10 \, a b x + 3 \, a^{2}\right )} e^{2} + 4 \, {\left (5 \, b^{2} d x + a b d\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(8*b^2*d^2 + (15*b^2*x^2 + 10*a*b*x + 3*a^2)*e^2 + 4*(5*b^2*d*x + a*b*d)*e)*sqrt(x*e + d)/(x^3*e^6 + 3*d
*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (63) = 126\).
time = 0.63, size = 389, normalized size = 5.64 \begin {gather*} \begin {cases} - \frac {6 a^{2} e^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {8 a b d e}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {20 a b e^{2} x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {16 b^{2} d^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {40 b^{2} d e x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {30 b^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a**2*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x))
- 8*a*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 20*a*b*e**
2*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*b**2*d**2/(15*d
**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*b**2*d*e*x/(15*d**2*e**3
*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*b**2*e**2*x**2/(15*d**2*e**3*sqr
t(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x + a*b*x**2 + b**2*x*
*3/3)/d**(7/2), True))

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Giac [A]
time = 0.81, size = 72, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} b^{2} - 10 \, {\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} + 10 \, {\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^2*b^2 - 10*(x*e + d)*b^2*d + 3*b^2*d^2 + 10*(x*e + d)*a*b*e - 6*a*b*d*e + 3*a^2*e^2)*e^(-3
)/(x*e + d)^(5/2)

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Mupad [B]
time = 0.57, size = 62, normalized size = 0.90 \begin {gather*} -\frac {6\,a^2\,e^2+8\,a\,b\,d\,e+20\,a\,b\,e^2\,x+16\,b^2\,d^2+40\,b^2\,d\,e\,x+30\,b^2\,e^2\,x^2}{15\,e^3\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)/(d + e*x)^(7/2),x)

[Out]

-(6*a^2*e^2 + 16*b^2*d^2 + 30*b^2*e^2*x^2 + 20*a*b*e^2*x + 40*b^2*d*e*x + 8*a*b*d*e)/(15*e^3*(d + e*x)^(5/2))

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