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

Optimal. Leaf size=115 \[ \frac{2 \sqrt{c+d x} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4}-\frac{2 (c+d x)^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^4}+\frac{2 (c+d x)^{5/2} (C d-3 c D)}{5 d^4}+\frac{2 D (c+d x)^{7/2}}{7 d^4} \]

[Out]

(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^4 - (2*(2*c*C*d - B*d^2 - 3*c^2*D)*(c + d*x)^(3/2))/(3
*d^4) + (2*(C*d - 3*c*D)*(c + d*x)^(5/2))/(5*d^4) + (2*D*(c + d*x)^(7/2))/(7*d^4)

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Rubi [A]  time = 0.0723718, antiderivative size = 115, 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} \[ \frac{2 \sqrt{c+d x} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4}-\frac{2 (c+d x)^{3/2} \left (-B d^2-3 c^2 D+2 c C d\right )}{3 d^4}+\frac{2 (c+d x)^{5/2} (C d-3 c D)}{5 d^4}+\frac{2 D (c+d x)^{7/2}}{7 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0903596, size = 82, normalized size = 0.71 \[ \frac{2 \sqrt{c+d x} \left (d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))+8 c^2 d (7 C+3 D x)-48 c^3 D\right )}{105 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B + x*(14*C + 9*D*x)) + d^3*(105*A + x*(35*B
+ 3*x*(7*C + 5*D*x)))))/(105*d^4)

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Maple [A]  time = 0.003, size = 91, normalized size = 0.8 \begin{align*}{\frac{30\,D{x}^{3}{d}^{3}+42\,C{d}^{3}{x}^{2}-36\,Dc{d}^{2}{x}^{2}+70\,B{d}^{3}x-56\,Cc{d}^{2}x+48\,D{c}^{2}dx+210\,A{d}^{3}-140\,Bc{d}^{2}+112\,C{c}^{2}d-96\,D{c}^{3}}{105\,{d}^{4}}\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)^(1/2),x)

[Out]

2/105*(d*x+c)^(1/2)*(15*D*d^3*x^3+21*C*d^3*x^2-18*D*c*d^2*x^2+35*B*d^3*x-28*C*c*d^2*x+24*D*c^2*d*x+105*A*d^3-7
0*B*c*d^2+56*C*c^2*d-48*D*c^3)/d^4

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Maxima [A]  time = 1.27153, size = 173, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (105 \, \sqrt{d x + c} A + \frac{35 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} B}{d} + \frac{7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} C}{d^{2}} + \frac{3 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{d x + c} c^{3}\right )} D}{d^{3}}\right )}}{105 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/105*(105*sqrt(d*x + c)*A + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x + c
)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C/d^2 + 3*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2
- 35*sqrt(d*x + c)*c^3)*D/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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 14.0802, size = 354, normalized size = 3.08 \begin{align*} \begin{cases} - \frac{\frac{2 A c}{\sqrt{c + d x}} + 2 A \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right ) + \frac{2 B c \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right )}{d} + \frac{2 B \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d} + \frac{2 C c \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d^{2}} + \frac{2 C \left (- \frac{c^{3}}{\sqrt{c + d x}} - 3 c^{2} \sqrt{c + d x} + c \left (c + d x\right )^{\frac{3}{2}} - \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} + \frac{2 D c \left (- \frac{c^{3}}{\sqrt{c + d x}} - 3 c^{2} \sqrt{c + d x} + c \left (c + d x\right )^{\frac{3}{2}} - \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{3}} + \frac{2 D \left (\frac{c^{4}}{\sqrt{c + d x}} + 4 c^{3} \sqrt{c + d x} - 2 c^{2} \left (c + d x\right )^{\frac{3}{2}} + \frac{4 c \left (c + d x\right )^{\frac{5}{2}}}{5} - \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{3}}}{d} & \text{for}\: d \neq 0 \\\frac{A x + \frac{B x^{2}}{2} + \frac{C x^{3}}{3} + \frac{D x^{4}}{4}}{\sqrt{c}} & \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)**(1/2),x)

[Out]

Piecewise((-(2*A*c/sqrt(c + d*x) + 2*A*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 2*B*c*(-c/sqrt(c + d*x) - sqrt(c +
d*x))/d + 2*B*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 2*C*c*(c**2/sqrt(c + d*x) + 2
*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 2*C*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(
3/2) - (c + d*x)**(5/2)/5)/d**2 + 2*D*c*(-c**3/sqrt(c + d*x) - 3*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c
+ d*x)**(5/2)/5)/d**3 + 2*D*(c**4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*
x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**3)/d, Ne(d, 0)), ((A*x + B*x**2/2 + C*x**3/3 + D*x**4/4)/sqrt(c), True))

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Giac [A]  time = 1.56262, size = 173, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (105 \, \sqrt{d x + c} A + \frac{35 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} B}{d} + \frac{7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c + 15 \, \sqrt{d x + c} c^{2}\right )} C}{d^{2}} + \frac{3 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{d x + c} c^{3}\right )} D}{d^{3}}\right )}}{105 \, 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)^(1/2),x, algorithm="giac")

[Out]

2/105*(105*sqrt(d*x + c)*A + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*B/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x + c
)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*C/d^2 + 3*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2
- 35*sqrt(d*x + c)*c^3)*D/d^3)/d