3.21 \(\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^4 (c+d x)^{3/2}}+\frac{2 \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 \sqrt{c+d x}}+\frac{2 \sqrt{c+d x} (C d-3 c D)}{d^4}+\frac{2 D (c+d x)^{3/2}}{3 d^4} \]

[Out]

(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^4*(c + d*x)^(3/2)) + (2*(2*c*C*d - B*d^2 - 3*c^2*D))/(d^4*Sqrt[c
+ d*x]) + (2*(C*d - 3*c*D)*Sqrt[c + d*x])/d^4 + (2*D*(c + d*x)^(3/2))/(3*d^4)

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Rubi [A]  time = 0.0721717, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1850} \[ -\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d^4 (c+d x)^{3/2}}+\frac{2 \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 \sqrt{c+d x}}+\frac{2 \sqrt{c+d x} (C d-3 c D)}{d^4}+\frac{2 D (c+d x)^{3/2}}{3 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(5/2),x]

[Out]

(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^4*(c + d*x)^(3/2)) + (2*(2*c*C*d - B*d^2 - 3*c^2*D))/(d^4*Sqrt[c
+ d*x]) + (2*(C*d - 3*c*D)*Sqrt[c + d*x])/d^4 + (2*D*(c + d*x)^(3/2))/(3*d^4)

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{A+B x+C x^2+D x^3}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{c^2 C d-B c d^2+A d^3-c^3 D}{d^3 (c+d x)^{5/2}}+\frac{-2 c C d+B d^2+3 c^2 D}{d^3 (c+d x)^{3/2}}+\frac{C d-3 c D}{d^3 \sqrt{c+d x}}+\frac{D \sqrt{c+d x}}{d^3}\right ) \, dx\\ &=-\frac{2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{3 d^4 (c+d x)^{3/2}}+\frac{2 \left (2 c C d-B d^2-3 c^2 D\right )}{d^4 \sqrt{c+d x}}+\frac{2 (C d-3 c D) \sqrt{c+d x}}{d^4}+\frac{2 D (c+d x)^{3/2}}{3 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0844059, size = 75, normalized size = 0.66 \[ -\frac{2 \left (d^3 \left (A+3 B x+x^2 (-(3 C+D x))\right )+2 c d^2 (B+3 x (D x-2 C))-8 c^2 d (C-3 D x)+16 c^3 D\right )}{3 d^4 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(5/2),x]

[Out]

(-2*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)) + d^3*(A + 3*B*x - x^2*(3*C + D*x))))/(3*
d^4*(c + d*x)^(3/2))

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Maple [A]  time = 0.004, size = 90, normalized size = 0.8 \begin{align*} -{\frac{-2\,D{x}^{3}{d}^{3}-6\,C{d}^{3}{x}^{2}+12\,Dc{d}^{2}{x}^{2}+6\,B{d}^{3}x-24\,Cc{d}^{2}x+48\,D{c}^{2}dx+2\,A{d}^{3}+4\,Bc{d}^{2}-16\,C{c}^{2}d+32\,D{c}^{3}}{3\,{d}^{4}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)

[Out]

-2/3/(d*x+c)^(3/2)*(-D*d^3*x^3-3*C*d^3*x^2+6*D*c*d^2*x^2+3*B*d^3*x-12*C*c*d^2*x+24*D*c^2*d*x+A*d^3+2*B*c*d^2-8
*C*c^2*d+16*D*c^3)/d^4

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Maxima [A]  time = 3.07524, size = 132, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{3}{2}} D - 3 \,{\left (3 \, D c - C d\right )} \sqrt{d x + c}}{d^{3}} + \frac{D c^{3} - C c^{2} d + B c d^{2} - A d^{3} - 3 \,{\left (3 \, D c^{2} - 2 \, C c d + B d^{2}\right )}{\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{3}}\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/3*(((d*x + c)^(3/2)*D - 3*(3*D*c - C*d)*sqrt(d*x + c))/d^3 + (D*c^3 - C*c^2*d + B*c*d^2 - A*d^3 - 3*(3*D*c^2
- 2*C*c*d + B*d^2)*(d*x + c))/((d*x + c)^(3/2)*d^3))/d

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 1.28978, size = 425, normalized size = 3.76 \begin{align*} \begin{cases} - \frac{2 A d^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{4 B c d^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{6 B d^{3} x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{16 C c^{2} d}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{24 C c d^{2} x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{6 C d^{3} x^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{32 D c^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{48 D c^{2} d x}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} - \frac{12 D c d^{2} x^{2}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} + \frac{2 D d^{3} x^{3}}{3 c d^{4} \sqrt{c + d x} + 3 d^{5} x \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{A x + \frac{B x^{2}}{2} + \frac{C x^{3}}{3} + \frac{D x^{4}}{4}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)

[Out]

Piecewise((-2*A*d**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 4*B*c*d**2/(3*c*d**4*sqrt(c + d*x) +
3*d**5*x*sqrt(c + d*x)) - 6*B*d**3*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 16*C*c**2*d/(3*c*d**4
*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 24*C*c*d**2*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 6
*C*d**3*x**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 32*D*c**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*
sqrt(c + d*x)) - 48*D*c**2*d*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 12*D*c*d**2*x**2/(3*c*d**4*
sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) + 2*D*d**3*x**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)), Ne(
d, 0)), ((A*x + B*x**2/2 + C*x**3/3 + D*x**4/4)/c**(5/2), True))

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Giac [A]  time = 2.48542, size = 155, normalized size = 1.37 \begin{align*} -\frac{2 \,{\left (9 \,{\left (d x + c\right )} D c^{2} - D c^{3} - 6 \,{\left (d x + c\right )} C c d + C c^{2} d + 3 \,{\left (d x + c\right )} B d^{2} - B c d^{2} + A d^{3}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} D d^{8} - 9 \, \sqrt{d x + c} D c d^{8} + 3 \, \sqrt{d x + c} C d^{9}\right )}}{3 \, d^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

-2/3*(9*(d*x + c)*D*c^2 - D*c^3 - 6*(d*x + c)*C*c*d + C*c^2*d + 3*(d*x + c)*B*d^2 - B*c*d^2 + A*d^3)/((d*x + c
)^(3/2)*d^4) + 2/3*((d*x + c)^(3/2)*D*d^8 - 9*sqrt(d*x + c)*D*c*d^8 + 3*sqrt(d*x + c)*C*d^9)/d^12