### 3.2281 $$\int \frac{(a+b x+c x^2)^2}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=162 $-\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}$

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^5*(d +
e*x)^(3/2)) - (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(e^5*Sqrt[d + e*x]) - (4*c*(2*c*d - b*e)*Sqrt[d
+ e*x])/e^5 + (2*c^2*(d + e*x)^(3/2))/(3*e^5)

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Rubi [A]  time = 0.0708952, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {698} $-\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^5*(d +
e*x)^(3/2)) - (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(e^5*Sqrt[d + e*x]) - (4*c*(2*c*d - b*e)*Sqrt[d
+ e*x])/e^5 + (2*c^2*(d + e*x)^(3/2))/(3*e^5)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.136638, size = 172, normalized size = 1.06 $-\frac{2 \left (3 a^2 e^4+2 a b e^3 (2 d+5 e x)+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(3*a^2*e^4 + 2*a*b*e^3*(2*d + 5*e*x) + b^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 2*a*c*e^2*(8*d^2 + 20*d*e
*x + 15*e^2*x^2) - 6*b*c*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + c^2*(128*d^4 + 320*d^3*e*x + 240
*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))

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Maple [A]  time = 0.044, size = 194, normalized size = 1.2 \begin{align*} -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}-60\,bc{e}^{4}{x}^{3}+80\,{c}^{2}d{e}^{3}{x}^{3}+60\,ac{e}^{4}{x}^{2}+30\,{b}^{2}{e}^{4}{x}^{2}-360\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+20\,ab{e}^{4}x+80\,acd{e}^{3}x+40\,{b}^{2}d{e}^{3}x-480\,bc{d}^{2}{e}^{2}x+640\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}+8\,abd{e}^{3}+32\,ac{d}^{2}{e}^{2}+16\,{b}^{2}{d}^{2}{e}^{2}-192\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/(e*x+d)^(5/2)*(-5*c^2*e^4*x^4-30*b*c*e^4*x^3+40*c^2*d*e^3*x^3+30*a*c*e^4*x^2+15*b^2*e^4*x^2-180*b*c*d*e^
3*x^2+240*c^2*d^2*e^2*x^2+10*a*b*e^4*x+40*a*c*d*e^3*x+20*b^2*d*e^3*x-240*b*c*d^2*e^2*x+320*c^2*d^3*e*x+3*a^2*e
^4+4*a*b*d*e^3+16*a*c*d^2*e^2+8*b^2*d^2*e^2-96*b*c*d^3*e+128*c^2*d^4)/e^5

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Maxima [A]  time = 0.964951, size = 250, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 6 \,{\left (2 \, c^{2} d - b c e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} - 6 \, b c d^{3} e - 6 \, a b d e^{3} + 3 \, a^{2} e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 15 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{2} - 10 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{4}}\right )}}{15 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.99409, size = 451, normalized size = 2.78 \begin{align*} \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 4 \, a b d e^{3} - 3 \, a^{2} e^{4} - 8 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} - 15 \,{\left (16 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 10 \,{\left (32 \, c^{2} d^{3} e - 24 \, b c d^{2} e^{2} + a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*c^2*e^4*x^4 - 128*c^2*d^4 + 96*b*c*d^3*e - 4*a*b*d*e^3 - 3*a^2*e^4 - 8*(b^2 + 2*a*c)*d^2*e^2 - 10*(4*c
^2*d*e^3 - 3*b*c*e^4)*x^3 - 15*(16*c^2*d^2*e^2 - 12*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 - 10*(32*c^2*d^3*e - 24
*b*c*d^2*e^2 + a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^
5)

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Sympy [A]  time = 3.81245, size = 1180, normalized size = 7.28 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*a**2*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 8*a*b*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 20*a*b*
e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 32*a*c*d**2*e**
2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*a*c*d*e**3*x/(15*
d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 60*a*c*e**4*x**2/(15*d**2*
e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 16*b**2*d**2*e**2/(15*d**2*e**5
*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 40*b**2*d*e**3*x/(15*d**2*e**5*sqrt
(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 30*b**2*e**4*x**2/(15*d**2*e**5*sqrt(d +
e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 192*b*c*d**3*e/(15*d**2*e**5*sqrt(d + e*x) +
30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 480*b*c*d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x) + 3
0*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 360*b*c*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*
d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 60*b*c*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**
6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*c**2*d**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt
(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*c**2*d**3*e*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d +
e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*c**2*d**2*e**2*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d
+ e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*c**2*d*e**3*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d +
e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*c**2*e**4*x**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x
) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4/2 + c*
*2*x**5/5)/d**(7/2), True))

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Giac [A]  time = 1.13767, size = 325, normalized size = 2.01 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10} + 6 \, \sqrt{x e + d} b c e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \,{\left (x e + d\right )}^{2} b c d e + 30 \,{\left (x e + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \,{\left (x e + d\right )}^{2} b^{2} e^{2} + 30 \,{\left (x e + d\right )}^{2} a c e^{2} - 10 \,{\left (x e + d\right )} b^{2} d e^{2} - 20 \,{\left (x e + d\right )} a c d e^{2} + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \,{\left (x e + d\right )} a b e^{3} - 6 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((x*e + d)^(3/2)*c^2*e^10 - 12*sqrt(x*e + d)*c^2*d*e^10 + 6*sqrt(x*e + d)*b*c*e^11)*e^(-15) - 2/15*(90*(x*
e + d)^2*c^2*d^2 - 20*(x*e + d)*c^2*d^3 + 3*c^2*d^4 - 90*(x*e + d)^2*b*c*d*e + 30*(x*e + d)*b*c*d^2*e - 6*b*c*
d^3*e + 15*(x*e + d)^2*b^2*e^2 + 30*(x*e + d)^2*a*c*e^2 - 10*(x*e + d)*b^2*d*e^2 - 20*(x*e + d)*a*c*d*e^2 + 3*
b^2*d^2*e^2 + 6*a*c*d^2*e^2 + 10*(x*e + d)*a*b*e^3 - 6*a*b*d*e^3 + 3*a^2*e^4)*e^(-5)/(x*e + d)^(5/2)