3.17.35 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{5/2}} \, dx\) [1635]

Optimal. Leaf size=125 \[ -\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {12 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \]

[Out]

-2/3*(-a*e+b*d)^4/e^5/(e*x+d)^(3/2)-8/3*b^3*(-a*e+b*d)*(e*x+d)^(3/2)/e^5+2/5*b^4*(e*x+d)^(5/2)/e^5+8*b*(-a*e+b
*d)^3/e^5/(e*x+d)^(1/2)+12*b^2*(-a*e+b*d)^2*(e*x+d)^(1/2)/e^5

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Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \begin {gather*} -\frac {8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac {12 b^2 \sqrt {d+e x} (b d-a e)^2}{e^5}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}-\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^4)/(3*e^5*(d + e*x)^(3/2)) + (8*b*(b*d - a*e)^3)/(e^5*Sqrt[d + e*x]) + (12*b^2*(b*d - a*e)^2*S
qrt[d + e*x])/e^5 - (8*b^3*(b*d - a*e)*(d + e*x)^(3/2))/(3*e^5) + (2*b^4*(d + e*x)^(5/2))/(5*e^5)

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 {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{5/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{3/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 \sqrt {d+e x}}-\frac {4 b^3 (b d-a e) \sqrt {d+e x}}{e^4}+\frac {b^4 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {12 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 153, normalized size = 1.22 \begin {gather*} \frac {2 \left (-5 a^4 e^4-20 a^3 b e^3 (2 d+3 e x)+30 a^2 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+20 a b^3 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^4 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-5*a^4*e^4 - 20*a^3*b*e^3*(2*d + 3*e*x) + 30*a^2*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 20*a*b^3*e*(-16*
d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^4*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*
x^4)))/(15*e^5*(d + e*x)^(3/2))

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Maple [A]
time = 0.67, size = 198, normalized size = 1.58

method	result	size
risch	\(\frac {2 b^{2} \left (3 b^{2} x^{2} e^{2}+20 a b \,e^{2} x -14 b^{2} d e x +90 a^{2} e^{2}-160 a b d e +73 b^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (12 b e x +a e +11 b d \right ) \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(128\)
gosper	\(-\frac {2 \left (-3 b^{4} x^{4} e^{4}-20 a \,b^{3} e^{4} x^{3}+8 b^{4} d \,e^{3} x^{3}-90 a^{2} b^{2} e^{4} x^{2}+120 a \,b^{3} d \,e^{3} x^{2}-48 b^{4} d^{2} e^{2} x^{2}+60 a^{3} b \,e^{4} x -360 a^{2} b^{2} d \,e^{3} x +480 a \,b^{3} d^{2} e^{2} x -192 b^{4} d^{3} e x +5 e^{4} a^{4}+40 a^{3} b d \,e^{3}-240 a^{2} b^{2} d^{2} e^{2}+320 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(186\)
trager	\(-\frac {2 \left (-3 b^{4} x^{4} e^{4}-20 a \,b^{3} e^{4} x^{3}+8 b^{4} d \,e^{3} x^{3}-90 a^{2} b^{2} e^{4} x^{2}+120 a \,b^{3} d \,e^{3} x^{2}-48 b^{4} d^{2} e^{2} x^{2}+60 a^{3} b \,e^{4} x -360 a^{2} b^{2} d \,e^{3} x +480 a \,b^{3} d^{2} e^{2} x -192 b^{4} d^{3} e x +5 e^{4} a^{4}+40 a^{3} b d \,e^{3}-240 a^{2} b^{2} d^{2} e^{2}+320 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(186\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a^{2} b^{2} e^{2} \sqrt {e x +d}-24 a \,b^{3} d e \sqrt {e x +d}+12 b^{4} d^{2} \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {8 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(198\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a^{2} b^{2} e^{2} \sqrt {e x +d}-24 a \,b^{3} d e \sqrt {e x +d}+12 b^{4} d^{2} \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {8 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^5*(1/5*b^4*(e*x+d)^(5/2)+4/3*a*b^3*e*(e*x+d)^(3/2)-4/3*b^4*d*(e*x+d)^(3/2)+6*a^2*b^2*e^2*(e*x+d)^(1/2)-12*
a*b^3*d*e*(e*x+d)^(1/2)+6*b^4*d^2*(e*x+d)^(1/2)-1/3*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4
*d^4)/(e*x+d)^(3/2)-4*b*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(e*x+d)^(1/2))

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Maxima [A]
time = 0.28, size = 187, normalized size = 1.50 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} - 20 \, {\left (b^{4} d - a b^{3} e\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 90 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {5 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4} - 12 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (x e + d\right )}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\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)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(x*e + d)^(5/2)*b^4 - 20*(b^4*d - a*b^3*e)*(x*e + d)^(3/2) + 90*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)
*sqrt(x*e + d))*e^(-4) - 5*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4 - 12*(b^4*d^
3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(x*e + d))*e^(-4)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 2.05, size = 183, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (128 \, b^{4} d^{4} + {\left (3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 90 \, a^{2} b^{2} x^{2} - 60 \, a^{3} b x - 5 \, a^{4}\right )} e^{4} - 8 \, {\left (b^{4} d x^{3} + 15 \, a b^{3} d x^{2} - 45 \, a^{2} b^{2} d x + 5 \, a^{3} b d\right )} e^{3} + 48 \, {\left (b^{4} d^{2} x^{2} - 10 \, a b^{3} d^{2} x + 5 \, a^{2} b^{2} d^{2}\right )} e^{2} + 64 \, {\left (3 \, b^{4} d^{3} x - 5 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\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)^2/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/15*(128*b^4*d^4 + (3*b^4*x^4 + 20*a*b^3*x^3 + 90*a^2*b^2*x^2 - 60*a^3*b*x - 5*a^4)*e^4 - 8*(b^4*d*x^3 + 15*a
*b^3*d*x^2 - 45*a^2*b^2*d*x + 5*a^3*b*d)*e^3 + 48*(b^4*d^2*x^2 - 10*a*b^3*d^2*x + 5*a^2*b^2*d^2)*e^2 + 64*(3*b
^4*d^3*x - 5*a*b^3*d^3)*e)*sqrt(x*e + d)/(x^2*e^7 + 2*d*x*e^6 + d^2*e^5)

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Sympy [A]
time = 18.44, size = 136, normalized size = 1.09 \begin {gather*} \frac {2 b^{4} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} - \frac {8 b \left (a e - b d\right )^{3}}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (8 a b^{3} e - 8 b^{4} d\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (12 a^{2} b^{2} e^{2} - 24 a b^{3} d e + 12 b^{4} d^{2}\right )}{e^{5}} - \frac {2 \left (a e - b d\right )^{4}}{3 e^{5} \left (d + e x\right )^{\frac {3}{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)**2/(e*x+d)**(5/2),x)

[Out]

2*b**4*(d + e*x)**(5/2)/(5*e**5) - 8*b*(a*e - b*d)**3/(e**5*sqrt(d + e*x)) + (d + e*x)**(3/2)*(8*a*b**3*e - 8*
b**4*d)/(3*e**5) + sqrt(d + e*x)*(12*a**2*b**2*e**2 - 24*a*b**3*d*e + 12*b**4*d**2)/e**5 - 2*(a*e - b*d)**4/(3
*e**5*(d + e*x)**(3/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (113) = 226\).
time = 1.13, size = 229, normalized size = 1.83 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d e^{20} + 90 \, \sqrt {x e + d} b^{4} d^{2} e^{20} + 20 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} e^{21} - 180 \, \sqrt {x e + d} a b^{3} d e^{21} + 90 \, \sqrt {x e + d} a^{2} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} b^{4} d^{3} - b^{4} d^{4} - 36 \, {\left (x e + d\right )} a b^{3} d^{2} e + 4 \, a b^{3} d^{3} e + 36 \, {\left (x e + d\right )} a^{2} b^{2} d e^{2} - 6 \, a^{2} b^{2} d^{2} e^{2} - 12 \, {\left (x e + d\right )} a^{3} b e^{3} + 4 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*b^4*e^20 - 20*(x*e + d)^(3/2)*b^4*d*e^20 + 90*sqrt(x*e + d)*b^4*d^2*e^20 + 20*(x*e + d
)^(3/2)*a*b^3*e^21 - 180*sqrt(x*e + d)*a*b^3*d*e^21 + 90*sqrt(x*e + d)*a^2*b^2*e^22)*e^(-25) + 2/3*(12*(x*e +
d)*b^4*d^3 - b^4*d^4 - 36*(x*e + d)*a*b^3*d^2*e + 4*a*b^3*d^3*e + 36*(x*e + d)*a^2*b^2*d*e^2 - 6*a^2*b^2*d^2*e
^2 - 12*(x*e + d)*a^3*b*e^3 + 4*a^3*b*d*e^3 - a^4*e^4)*e^(-5)/(x*e + d)^(3/2)

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Mupad [B]
time = 0.07, size = 175, normalized size = 1.40 \begin {gather*} \frac {\left (d+e\,x\right )\,\left (-8\,a^3\,b\,e^3+24\,a^2\,b^2\,d\,e^2-24\,a\,b^3\,d^2\,e+8\,b^4\,d^3\right )-\frac {2\,a^4\,e^4}{3}-\frac {2\,b^4\,d^4}{3}-4\,a^2\,b^2\,d^2\,e^2+\frac {8\,a\,b^3\,d^3\,e}{3}+\frac {8\,a^3\,b\,d\,e^3}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,b^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}}{e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)*(8*b^4*d^3 - 8*a^3*b*e^3 + 24*a^2*b^2*d*e^2 - 24*a*b^3*d^2*e) - (2*a^4*e^4)/3 - (2*b^4*d^4)/3 - 4*a
^2*b^2*d^2*e^2 + (8*a*b^3*d^3*e)/3 + (8*a^3*b*d*e^3)/3)/(e^5*(d + e*x)^(3/2)) + (2*b^4*(d + e*x)^(5/2))/(5*e^5
) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(3/2))/(3*e^5) + (12*b^2*(a*e - b*d)^2*(d + e*x)^(1/2))/e^5

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