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

Optimal. Leaf size=113 \[ -\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4} \]

[Out]

(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^4*Sqrt[c + d*x]) - (2*(2*c*C*d - B*d^2 - 3*c^2*D)*Sqrt[c + d*x])/d
^4 + (2*(C*d - 3*c*D)*(c + d*x)^(3/2))/(3*d^4) + (2*D*(c + d*x)^(5/2))/(5*d^4)

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Rubi [A]  time = 0.0810817, 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} \[ -\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0800003, size = 82, normalized size = 0.73 \[ \frac{2 \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))-8 c^2 d (5 C-3 D x)+48 c^3 D\right )}{15 d^4 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(-15*A + x*(15*B + 5*C*x + 3*D*
x^2))))/(15*d^4*Sqrt[c + d*x])

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Maple [A]  time = 0.004, size = 91, normalized size = 0.8 \begin{align*} -{\frac{-6\,D{x}^{3}{d}^{3}-10\,C{d}^{3}{x}^{2}+12\,Dc{d}^{2}{x}^{2}-30\,B{d}^{3}x+40\,Cc{d}^{2}x-48\,D{c}^{2}dx+30\,A{d}^{3}-60\,Bc{d}^{2}+80\,C{c}^{2}d-96\,D{c}^{3}}{15\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \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)^(3/2),x)

[Out]

-2/15/(d*x+c)^(1/2)*(-3*D*d^3*x^3-5*C*d^3*x^2+6*D*c*d^2*x^2-15*B*d^3*x+20*C*c*d^2*x-24*D*c^2*d*x+15*A*d^3-30*B
*c*d^2+40*C*c^2*d-48*D*c^3)/d^4

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Maxima [A]  time = 3.07117, size = 138, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} D - 5 \,{\left (3 \, D c - C d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, D c^{2} - 2 \, C c d + B d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{15 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{15 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/15*((3*(d*x + c)^(5/2)*D - 5*(3*D*c - C*d)*(d*x + c)^(3/2) + 15*(3*D*c^2 - 2*C*c*d + B*d^2)*sqrt(d*x + c))/d
^3 + 15*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)/(sqrt(d*x + c)*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)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 14.2075, size = 114, normalized size = 1.01 \begin{align*} \frac{2 D \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 C d - 6 D c\right )}{3 d^{4}} + \frac{\sqrt{c + d x} \left (2 B d^{2} - 4 C c d + 6 D c^{2}\right )}{d^{4}} + \frac{2 \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{4} \sqrt{c + d x}} \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)**(3/2),x)

[Out]

2*D*(c + d*x)**(5/2)/(5*d**4) + (c + d*x)**(3/2)*(2*C*d - 6*D*c)/(3*d**4) + sqrt(c + d*x)*(2*B*d**2 - 4*C*c*d
+ 6*D*c**2)/d**4 + 2*(-A*d**3 + B*c*d**2 - C*c**2*d + D*c**3)/(d**4*sqrt(c + d*x))

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Giac [A]  time = 2.30635, size = 171, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} D d^{16} - 15 \,{\left (d x + c\right )}^{\frac{3}{2}} D c d^{16} + 45 \, \sqrt{d x + c} D c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C d^{17} - 30 \, \sqrt{d x + c} C c d^{17} + 15 \, \sqrt{d x + c} B d^{18}\right )}}{15 \, d^{20}} \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)^(3/2),x, algorithm="giac")

[Out]

2*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)/(sqrt(d*x + c)*d^4) + 2/15*(3*(d*x + c)^(5/2)*D*d^16 - 15*(d*x + c)^(3/2
)*D*c*d^16 + 45*sqrt(d*x + c)*D*c^2*d^16 + 5*(d*x + c)^(3/2)*C*d^17 - 30*sqrt(d*x + c)*C*c*d^17 + 15*sqrt(d*x
+ c)*B*d^18)/d^20