### 3.352 $$\int \frac{(b x+c x^2)^2}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=143 $-\frac{2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}$

[Out]

(-2*d^2*(c*d - b*e)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))/(3*e^5*(d + e*x)^(3/2)) - (2*
(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2))/(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 = 0.0576748, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.048, Rules used = {698} $-\frac{2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*d^2*(c*d - b*e)^2)/(5*e^5*(d + e*x)^(5/2)) + (4*d*(c*d - b*e)*(2*c*d - b*e))/(3*e^5*(d + e*x)^(3/2)) - (2*
(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2))/(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 (b x+c x^2\right )^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^{7/2}}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^{5/2}}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{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 d^2 (c d-b e)^2}{5 e^5 (d+e x)^{5/2}}+\frac{4 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\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 = 0.0663874, size = 123, normalized size = 0.86 $-\frac{2 \left (b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(b^2*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 = 0.048, size = 141, normalized size = 1. \begin{align*} -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}-60\,bc{e}^{4}{x}^{3}+80\,{c}^{2}d{e}^{3}{x}^{3}+30\,{b}^{2}{e}^{4}{x}^{2}-360\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+40\,{b}^{2}d{e}^{3}x-480\,bc{d}^{2}{e}^{2}x+640\,{c}^{2}{d}^{3}ex+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)^2/(e*x+d)^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.16068, size = 198, normalized size = 1.38 \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 + 3 \, b^{2} d^{2} e^{2} + 15 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 10 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} 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)^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 + 3*b^2*d^2*e
^2 + 15*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(e*x + d)^2 - 10*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d))/((
e*x + d)^(5/2)*e^4))/e

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Fricas [A]  time = 2.00331, size = 359, normalized size = 2.51 \begin{align*} \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 8 \, b^{2} 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} + b^{2} e^{4}\right )} x^{2} - 20 \,{\left (16 \, c^{2} d^{3} e - 12 \, b c d^{2} e^{2} + b^{2} 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)^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 - 8*b^2*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*e^4)*x^2 - 20*(16*c^2*d^3*e - 12*b*c*d^2*e^2 + b^2*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 = 4.84365, size = 787, normalized size = 5.5 \begin{align*} \begin{cases} - \frac{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}} - \frac{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}} - \frac{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}} + \frac{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}} + \frac{480 b c d^{2} e^{2} 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}} + \frac{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}} + \frac{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}} - \frac{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}} - \frac{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}} - \frac{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}} - \frac{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}} + \frac{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}} & \text{for}\: e \neq 0 \\\frac{\frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-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)) + 1
92*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) + 30*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)), ((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 = 1.34746, size = 242, normalized size = 1.69 \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} - 10 \,{\left (x e + d\right )} b^{2} d e^{2} + 3 \, b^{2} d^{2} e^{2}\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)^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 - 10*(x*e + d)*b^2*d*e^2 + 3*b^2*d^2*e^2)*e^(-5)/(x*e + d)^(5/2)