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

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

[Out]

2/3*(C*d-3*D*c)*(d*x+c)^(3/2)/d^4+2/5*D*(d*x+c)^(5/2)/d^4-2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^4/(d*x+c)^(1/2)-2*
(-B*d^2+2*C*c*d-3*D*c^2)*(d*x+c)^(1/2)/d^4

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Rubi [A]  time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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*}

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Mathematica [A]  time = 0.09, size = 82, normalized size = 0.73 \[ \frac {2 \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))+48 c^3 D-8 c^2 d (5 C-3 D x)\right )}{15 d^4 \sqrt {c+d x}} \]

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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fricas [A]  time = 0.81, size = 100, normalized size = 0.88 \[ \frac {2 \, {\left (3 \, D d^{3} x^{3} + 48 \, D c^{3} - 40 \, C c^{2} d + 30 \, B c d^{2} - 15 \, A d^{3} - {\left (6 \, D c d^{2} - 5 \, C d^{3}\right )} x^{2} + {\left (24 \, D c^{2} d - 20 \, C c d^{2} + 15 \, B d^{3}\right )} x\right )} \sqrt {d x + c}}{15 \, {\left (d^{5} x + c d^{4}\right )}} \]

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]

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

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giac [A]  time = 1.29, size = 127, normalized size = 1.12 \[ \frac {2 \, {\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt {d x + c} d^{4}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} D d^{16} - 15 \, {\left (d x + c\right )}^{\frac {3}{2}} D c d^{16} + 45 \, \sqrt {d x + c} D c^{2} d^{16} + 5 \, {\left (d x + c\right )}^{\frac {3}{2}} C d^{17} - 30 \, \sqrt {d x + c} C c d^{17} + 15 \, \sqrt {d x + c} B d^{18}\right )}}{15 \, d^{20}} \]

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

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maple [A]  time = 0.00, size = 91, normalized size = 0.81 \[ -\frac {2 \left (-3 D x^{3} d^{3}-5 C \,d^{3} x^{2}+6 D c \,d^{2} x^{2}-15 B \,d^{3} x +20 C c \,d^{2} x -24 D c^{2} d x +15 A \,d^{3}-30 B c \,d^{2}+40 C \,c^{2} d -48 D c^{3}\right )}{15 \sqrt {d x +c}\, d^{4}} \]

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 = 0.43, size = 102, normalized size = 0.90 \[ \frac {2 \, {\left (\frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} D - 5 \, {\left (3 \, D c - C d\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, D c^{2} - 2 \, C c d + B d^{2}\right )} \sqrt {d x + c}}{d^{3}} + \frac {15 \, {\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt {d x + c} d^{3}}\right )}}{15 \, d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (c+d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 20.14, size = 114, normalized size = 1.01 \[ \frac {2 D \left (c + d x\right )^{\frac {5}{2}}}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (2 C d - 6 D c\right )}{3 d^{4}} + \frac {\sqrt {c + d x} \left (2 B d^{2} - 4 C c d + 6 D c^{2}\right )}{d^{4}} + \frac {2 \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{4} \sqrt {c + d x}} \]

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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