### 3.2279 $$\int \frac{(a+b x+c x^2)^2}{(d+e x)^{3/2}} \, dx$$

Optimal. Leaf size=162 $\frac{2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}$

[Out]

(-2*(c*d^2 - b*d*e + a*e^2)^2)/(e^5*Sqrt[d + e*x]) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x])/e
^5 + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(3/2))/(3*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(5
/2))/(5*e^5) + (2*c^2*(d + e*x)^(7/2))/(7*e^5)

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Rubi [A]  time = 0.0700174, 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} $\frac{2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac{4 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.134387, size = 171, normalized size = 1.06 $\frac{-70 e^2 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )+28 c e \left (5 a e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 b \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )-6 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )}{105 e^5 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-6*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) - 70*e^2*(3*a^2*e^2 - 6*a*b*e*(2*d +
e*x) + b^2*(8*d^2 + 4*d*e*x - e^2*x^2)) + 28*c*e*(5*a*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + 3*b*(16*d^3 + 8*d^2*e*
x - 2*d*e^2*x^2 + e^3*x^3)))/(105*e^5*Sqrt[d + e*x])

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Maple [A]  time = 0.045, size = 194, normalized size = 1.2 \begin{align*} -{\frac{-30\,{c}^{2}{x}^{4}{e}^{4}-84\,bc{e}^{4}{x}^{3}+48\,{c}^{2}d{e}^{3}{x}^{3}-140\,ac{e}^{4}{x}^{2}-70\,{b}^{2}{e}^{4}{x}^{2}+168\,bcd{e}^{3}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-420\,ab{e}^{4}x+560\,acd{e}^{3}x+280\,{b}^{2}d{e}^{3}x-672\,bc{d}^{2}{e}^{2}x+384\,{c}^{2}{d}^{3}ex+210\,{a}^{2}{e}^{4}-840\,abd{e}^{3}+1120\,ac{d}^{2}{e}^{2}+560\,{b}^{2}{d}^{2}{e}^{2}-1344\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{105\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(e*x+d)^(3/2),x)

[Out]

-2/105/(e*x+d)^(1/2)*(-15*c^2*e^4*x^4-42*b*c*e^4*x^3+24*c^2*d*e^3*x^3-70*a*c*e^4*x^2-35*b^2*e^4*x^2+84*b*c*d*e
^3*x^2-48*c^2*d^2*e^2*x^2-210*a*b*e^4*x+280*a*c*d*e^3*x+140*b^2*d*e^3*x-336*b*c*d^2*e^2*x+192*c^2*d^3*e*x+105*
a^2*e^4-420*a*b*d*e^3+560*a*c*d^2*e^2+280*b^2*d^2*e^2-672*b*c*d^3*e+384*c^2*d^4)/e^5

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Maxima [A]  time = 0.971692, size = 248, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 42 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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 )}^{\frac{3}{2}} - 210 \,{\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 )} \sqrt{e x + d}}{e^{4}} - \frac{105 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{105 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*(e*x + d)^(7/2)*c^2 - 42*(2*c^2*d - b*c*e)*(e*x + d)^(5/2) + 35*(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a
*c)*e^2)*(e*x + d)^(3/2) - 210*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*sqrt(e*x + d))/e^4 -
105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)/(sqrt(e*x + d)*e^4))/e

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Fricas [A]  time = 2.32187, size = 425, normalized size = 2.62 \begin{align*} \frac{2 \,{\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e + 420 \, a b d e^{3} - 105 \, a^{2} e^{4} - 280 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \,{\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} +{\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \,{\left (96 \, c^{2} d^{3} e - 168 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 70 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{6} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/105*(15*c^2*e^4*x^4 - 384*c^2*d^4 + 672*b*c*d^3*e + 420*a*b*d*e^3 - 105*a^2*e^4 - 280*(b^2 + 2*a*c)*d^2*e^2
- 6*(4*c^2*d*e^3 - 7*b*c*e^4)*x^3 + (48*c^2*d^2*e^2 - 84*b*c*d*e^3 + 35*(b^2 + 2*a*c)*e^4)*x^2 - 2*(96*c^2*d^3
*e - 168*b*c*d^2*e^2 - 105*a*b*e^4 + 70*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

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Sympy [A]  time = 29.9527, size = 182, normalized size = 1.12 \begin{align*} \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (4 b c e - 8 c^{2} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 a c e^{2} + 2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (4 a b e^{3} - 8 a c d e^{2} - 4 b^{2} d e^{2} + 12 b c d^{2} e - 8 c^{2} d^{3}\right )}{e^{5}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5} \sqrt{d + e x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.11763, size = 342, normalized size = 2.11 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} e^{30} - 84 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d e^{30} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt{x e + d} c^{2} d^{3} e^{30} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} b c e^{31} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b c d e^{31} + 630 \, \sqrt{x e + d} b c d^{2} e^{31} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{32} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c e^{32} - 210 \, \sqrt{x e + d} b^{2} d e^{32} - 420 \, \sqrt{x e + d} a c d e^{32} + 210 \, \sqrt{x e + d} a b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*e^30 - 84*(x*e + d)^(5/2)*c^2*d*e^30 + 210*(x*e + d)^(3/2)*c^2*d^2*e^30 - 420*sq
rt(x*e + d)*c^2*d^3*e^30 + 42*(x*e + d)^(5/2)*b*c*e^31 - 210*(x*e + d)^(3/2)*b*c*d*e^31 + 630*sqrt(x*e + d)*b*
c*d^2*e^31 + 35*(x*e + d)^(3/2)*b^2*e^32 + 70*(x*e + d)^(3/2)*a*c*e^32 - 210*sqrt(x*e + d)*b^2*d*e^32 - 420*sq
rt(x*e + d)*a*c*d*e^32 + 210*sqrt(x*e + d)*a*b*e^33)*e^(-35) - 2*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*
d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*e^(-5)/sqrt(x*e + d)