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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, 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, 43} \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 43

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*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 101, normalized size = 0.81 \begin {gather*} \frac {2 \left (-20 b^3 (d+e x)^3 (b d-a e)+90 b^2 (d+e x)^2 (b d-a e)^2+60 b (d+e x) (b d-a e)^3-5 (b d-a e)^4+3 b^4 (d+e x)^4\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*(b*d - a*e)^4 + 60*b*(b*d - a*e)^3*(d + e*x) + 90*b^2*(b*d - a*e)^2*(d + e*x)^2 - 20*b^3*(b*d - a*e)*(d
+ e*x)^3 + 3*b^4*(d + e*x)^4))/(15*e^5*(d + e*x)^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.09, size = 213, normalized size = 1.70 \begin {gather*} \frac {2 \left (-5 a^4 e^4-60 a^3 b e^3 (d+e x)+20 a^3 b d e^3-30 a^2 b^2 d^2 e^2+90 a^2 b^2 e^2 (d+e x)^2+180 a^2 b^2 d e^2 (d+e x)+20 a b^3 d^3 e-180 a b^3 d^2 e (d+e x)+20 a b^3 e (d+e x)^3-180 a b^3 d e (d+e x)^2-5 b^4 d^4+60 b^4 d^3 (d+e x)+90 b^4 d^2 (d+e x)^2+3 b^4 (d+e x)^4-20 b^4 d (d+e x)^3\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-5*b^4*d^4 + 20*a*b^3*d^3*e - 30*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 5*a^4*e^4 + 60*b^4*d^3*(d + e*x) - 180
*a*b^3*d^2*e*(d + e*x) + 180*a^2*b^2*d*e^2*(d + e*x) - 60*a^3*b*e^3*(d + e*x) + 90*b^4*d^2*(d + e*x)^2 - 180*a
*b^3*d*e*(d + e*x)^2 + 90*a^2*b^2*e^2*(d + e*x)^2 - 20*b^4*d*(d + e*x)^3 + 20*a*b^3*e*(d + e*x)^3 + 3*b^4*(d +
e*x)^4))/(15*e^5*(d + e*x)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 203, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (3 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 320 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} - 40 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} - 4 \, {\left (2 \, b^{4} d e^{3} - 5 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{2} e^{2} - 20 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 12 \, {\left (16 \, b^{4} d^{3} e - 40 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} - 5 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + 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*(3*b^4*e^4*x^4 + 128*b^4*d^4 - 320*a*b^3*d^3*e + 240*a^2*b^2*d^2*e^2 - 40*a^3*b*d*e^3 - 5*a^4*e^4 - 4*(2*
b^4*d*e^3 - 5*a*b^3*e^4)*x^3 + 6*(8*b^4*d^2*e^2 - 20*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 12*(16*b^4*d^3*e - 40
*a*b^3*d^2*e^2 + 30*a^2*b^2*d*e^3 - 5*a^3*b*e^4)*x)*sqrt(e*x + d)/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

________________________________________________________________________________________

giac [B]  time = 0.20, 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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 186, normalized size = 1.49 \begin {gather*} -\frac {2 \left (-3 b^{4} e^{4} x^{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 a^{4} e^{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}} \end {gather*}

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)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.02, size = 187, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} - 20 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + 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 {e x + d}}{e^{4}} - \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 (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{4}}\right )}}{15 \, e} \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*(e*x + d)^(5/2)*b^4 - 20*(b^4*d - a*b^3*e)*(e*x + d)^(3/2) + 90*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)
*sqrt(e*x + 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)*(e*x + d))/((e*x + d)^(3/2)*e^4))/e

________________________________________________________________________________________

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

________________________________________________________________________________________

sympy [A]  time = 45.31, 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}} \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))

________________________________________________________________________________________