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

Optimal. Leaf size=123 \[ -\frac {8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac {4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac {8 b \sqrt {d+e x} (b d-a e)^3}{e^5}-\frac {2 (b d-a e)^4}{e^5 \sqrt {d+e x}}+\frac {2 b^4 (d+e x)^{7/2}}{7 e^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 123, 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)^{5/2} (b d-a e)}{5 e^5}+\frac {4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac {8 b \sqrt {d+e x} (b d-a e)^3}{e^5}-\frac {2 (b d-a e)^4}{e^5 \sqrt {d+e x}}+\frac {2 b^4 (d+e x)^{7/2}}{7 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^4)/(e^5*Sqrt[d + e*x]) - (8*b*(b*d - a*e)^3*Sqrt[d + e*x])/e^5 + (4*b^2*(b*d - a*e)^2*(d + e*x
)^(3/2))/e^5 - (8*b^3*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^5) + (2*b^4*(d + e*x)^(7/2))/(7*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)^{3/2}} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{3/2}}-\frac {4 b (b d-a e)^3}{e^4 \sqrt {d+e x}}+\frac {6 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac {b^4 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac {2 (b d-a e)^4}{e^5 \sqrt {d+e x}}-\frac {8 b (b d-a e)^3 \sqrt {d+e x}}{e^5}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{5/2}}{5 e^5}+\frac {2 b^4 (d+e x)^{7/2}}{7 e^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 101, normalized size = 0.82 \begin {gather*} \frac {2 \left (-28 b^3 (d+e x)^3 (b d-a e)+70 b^2 (d+e x)^2 (b d-a e)^2-140 b (d+e x) (b d-a e)^3-35 (b d-a e)^4+5 b^4 (d+e x)^4\right )}{35 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*(b*d - a*e)^4 - 140*b*(b*d - a*e)^3*(d + e*x) + 70*b^2*(b*d - a*e)^2*(d + e*x)^2 - 28*b^3*(b*d - a*e)*
(d + e*x)^3 + 5*b^4*(d + e*x)^4))/(35*e^5*Sqrt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.08, size = 213, normalized size = 1.73 \begin {gather*} \frac {2 \left (-35 a^4 e^4+140 a^3 b e^3 (d+e x)+140 a^3 b d e^3-210 a^2 b^2 d^2 e^2+70 a^2 b^2 e^2 (d+e x)^2-420 a^2 b^2 d e^2 (d+e x)+140 a b^3 d^3 e+420 a b^3 d^2 e (d+e x)+28 a b^3 e (d+e x)^3-140 a b^3 d e (d+e x)^2-35 b^4 d^4-140 b^4 d^3 (d+e x)+70 b^4 d^2 (d+e x)^2+5 b^4 (d+e x)^4-28 b^4 d (d+e x)^3\right )}{35 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*b^4*d^4 + 140*a*b^3*d^3*e - 210*a^2*b^2*d^2*e^2 + 140*a^3*b*d*e^3 - 35*a^4*e^4 - 140*b^4*d^3*(d + e*x)
+ 420*a*b^3*d^2*e*(d + e*x) - 420*a^2*b^2*d*e^2*(d + e*x) + 140*a^3*b*e^3*(d + e*x) + 70*b^4*d^2*(d + e*x)^2
- 140*a*b^3*d*e*(d + e*x)^2 + 70*a^2*b^2*e^2*(d + e*x)^2 - 28*b^4*d*(d + e*x)^3 + 28*a*b^3*e*(d + e*x)^3 + 5*b
^4*(d + e*x)^4))/(35*e^5*Sqrt[d + e*x])

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 192, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \, {\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \, {\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{6} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b^4*e^4*x^4 - 128*b^4*d^4 + 448*a*b^3*d^3*e - 560*a^2*b^2*d^2*e^2 + 280*a^3*b*d*e^3 - 35*a^4*e^4 - 4*(
2*b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 2*(8*b^4*d^2*e^2 - 28*a*b^3*d*e^3 + 35*a^2*b^2*e^4)*x^2 - 4*(16*b^4*d^3*e - 5
6*a*b^3*d^2*e^2 + 70*a^2*b^2*d*e^3 - 35*a^3*b*e^4)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

________________________________________________________________________________________

giac [B]  time = 0.19, size = 237, normalized size = 1.93 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} e^{30} - 28 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d e^{30} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{30} - 140 \, \sqrt {x e + d} b^{4} d^{3} e^{30} + 28 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} e^{31} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d e^{31} + 420 \, \sqrt {x e + d} a b^{3} d^{2} e^{31} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{32} - 420 \, \sqrt {x e + d} a^{2} b^{2} d e^{32} + 140 \, \sqrt {x e + d} a^{3} b e^{33}\right )} e^{\left (-35\right )} - \frac {2 \, {\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}\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^4*e^30 - 28*(x*e + d)^(5/2)*b^4*d*e^30 + 70*(x*e + d)^(3/2)*b^4*d^2*e^30 - 140*sqrt(
x*e + d)*b^4*d^3*e^30 + 28*(x*e + d)^(5/2)*a*b^3*e^31 - 140*(x*e + d)^(3/2)*a*b^3*d*e^31 + 420*sqrt(x*e + d)*a
*b^3*d^2*e^31 + 70*(x*e + d)^(3/2)*a^2*b^2*e^32 - 420*sqrt(x*e + d)*a^2*b^2*d*e^32 + 140*sqrt(x*e + d)*a^3*b*e
^33)*e^(-35) - 2*(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)*e^(-5)/sqrt(x*e + d)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 186, normalized size = 1.51 \begin {gather*} -\frac {2 \left (-5 b^{4} e^{4} x^{4}-28 a \,b^{3} e^{4} x^{3}+8 b^{4} d \,e^{3} x^{3}-70 a^{2} b^{2} e^{4} x^{2}+56 a \,b^{3} d \,e^{3} x^{2}-16 b^{4} d^{2} e^{2} x^{2}-140 a^{3} b \,e^{4} x +280 a^{2} b^{2} d \,e^{3} x -224 a \,b^{3} d^{2} e^{2} x +64 b^{4} d^{3} e x +35 a^{4} e^{4}-280 a^{3} b d \,e^{3}+560 a^{2} b^{2} d^{2} e^{2}-448 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{35 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/35*(-5*b^4*e^4*x^4-28*a*b^3*e^4*x^3+8*b^4*d*e^3*x^3-70*a^2*b^2*e^4*x^2+56*a*b^3*d*e^3*x^2-16*b^4*d^2*e^2*x^
2-140*a^3*b*e^4*x+280*a^2*b^2*d*e^3*x-224*a*b^3*d^2*e^2*x+64*b^4*d^3*e*x+35*a^4*e^4-280*a^3*b*d*e^3+560*a^2*b^
2*d^2*e^2-448*a*b^3*d^3*e+128*b^4*d^4)/(e*x+d)^(1/2)/e^5

________________________________________________________________________________________

maxima [A]  time = 1.03, size = 189, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} - 28 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 70 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 140 \, {\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 )} \sqrt {e x + d}}{e^{4}} - \frac {35 \, {\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}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{35 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*b^4 - 28*(b^4*d - a*b^3*e)*(e*x + d)^(5/2) + 70*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)
*(e*x + d)^(3/2) - 140*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*sqrt(e*x + d))/e^4 - 35*(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)/(sqrt(e*x + d)*e^4))/e

________________________________________________________________________________________

mupad [B]  time = 0.54, size = 153, normalized size = 1.24 \begin {gather*} \frac {2\,b^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {2\,a^4\,e^4-8\,a^3\,b\,d\,e^3+12\,a^2\,b^2\,d^2\,e^2-8\,a\,b^3\,d^3\,e+2\,b^4\,d^4}{e^5\,\sqrt {d+e\,x}}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,\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)^(3/2),x)

[Out]

(2*b^4*(d + e*x)^(7/2))/(7*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(5/2))/(5*e^5) - (2*a^4*e^4 + 2*b^4*d^4 + 1
2*a^2*b^2*d^2*e^2 - 8*a*b^3*d^3*e - 8*a^3*b*d*e^3)/(e^5*(d + e*x)^(1/2)) + (4*b^2*(a*e - b*d)^2*(d + e*x)^(3/2
))/e^5 + (8*b*(a*e - b*d)^3*(d + e*x)^(1/2))/e^5

________________________________________________________________________________________

sympy [A]  time = 33.85, size = 168, normalized size = 1.37 \begin {gather*} \frac {2 b^{4} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (8 a b^{3} e - 8 b^{4} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (12 a^{2} b^{2} e^{2} - 24 a b^{3} d e + 12 b^{4} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \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 )}{e^{5}} - \frac {2 \left (a e - b d\right )^{4}}{e^{5} \sqrt {d + e x}} \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)**(3/2),x)

[Out]

2*b**4*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(5/2)*(8*a*b**3*e - 8*b**4*d)/(5*e**5) + (d + e*x)**(3/2)*(12*a*
*2*b**2*e**2 - 24*a*b**3*d*e + 12*b**4*d**2)/(3*e**5) + sqrt(d + e*x)*(8*a**3*b*e**3 - 24*a**2*b**2*d*e**2 + 2
4*a*b**3*d**2*e - 8*b**4*d**3)/e**5 - 2*(a*e - b*d)**4/(e**5*sqrt(d + e*x))

________________________________________________________________________________________