3.4 \(\int \frac{c+d x+e x^2+f x^3}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 \sqrt{a+b x} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}+\frac{2 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{2 (a+b x)^{5/2} (b e-3 a f)}{5 b^4}+\frac{2 f (a+b x)^{7/2}}{7 b^4} \]

[Out]

(2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x])/b^4 + (2*(b^2*d - 2*a*b*e + 3*a^2*f)*(a + b*x)^(3/2))/(3
*b^4) + (2*(b*e - 3*a*f)*(a + b*x)^(5/2))/(5*b^4) + (2*f*(a + b*x)^(7/2))/(7*b^4)

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Rubi [A]  time = 0.0707695, antiderivative size = 114, 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} \[ \frac{2 \sqrt{a+b x} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}+\frac{2 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{2 (a+b x)^{5/2} (b e-3 a f)}{5 b^4}+\frac{2 f (a+b x)^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3)/Sqrt[a + b*x],x]

[Out]

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

Mathematica [A]  time = 0.125911, size = 82, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \left (8 a^2 b (7 e+3 f x)-48 a^3 f-2 a b^2 (35 d+x (14 e+9 f x))+b^3 (105 c+x (35 d+3 x (7 e+5 f x)))\right )}{105 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3)/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(-48*a^3*f + 8*a^2*b*(7*e + 3*f*x) - 2*a*b^2*(35*d + x*(14*e + 9*f*x)) + b^3*(105*c + x*(35*d
+ 3*x*(7*e + 5*f*x)))))/(105*b^4)

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Maple [A]  time = 0.045, size = 91, normalized size = 0.8 \begin{align*} -{\frac{-30\,f{x}^{3}{b}^{3}+36\,a{b}^{2}f{x}^{2}-42\,{b}^{3}e{x}^{2}-48\,{a}^{2}bfx+56\,a{b}^{2}ex-70\,{b}^{3}dx+96\,{a}^{3}f-112\,{a}^{2}be+140\,a{b}^{2}d-210\,{b}^{3}c}{105\,{b}^{4}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x)

[Out]

-2/105*(b*x+a)^(1/2)*(-15*b^3*f*x^3+18*a*b^2*f*x^2-21*b^3*e*x^2-24*a^2*b*f*x+28*a*b^2*e*x-35*b^3*d*x+48*a^3*f-
56*a^2*b*e+70*a*b^2*d-105*b^3*c)/b^4

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Maxima [A]  time = 0.961995, size = 173, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (105 \, \sqrt{b x + a} c + \frac{35 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} d}{b} + \frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} e}{b^{2}} + \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )} f}{b^{3}}\right )}}{105 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/105*(105*sqrt(b*x + a)*c + 35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*d/b + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a
)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2 + 3*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2
- 35*sqrt(b*x + a)*a^3)*f/b^3)/b

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Fricas [A]  time = 1.19074, size = 216, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (15 \, b^{3} f x^{3} + 105 \, b^{3} c - 70 \, a b^{2} d + 56 \, a^{2} b e - 48 \, a^{3} f + 3 \,{\left (7 \, b^{3} e - 6 \, a b^{2} f\right )} x^{2} +{\left (35 \, b^{3} d - 28 \, a b^{2} e + 24 \, a^{2} b f\right )} x\right )} \sqrt{b x + a}}{105 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*b^3*f*x^3 + 105*b^3*c - 70*a*b^2*d + 56*a^2*b*e - 48*a^3*f + 3*(7*b^3*e - 6*a*b^2*f)*x^2 + (35*b^3*d
- 28*a*b^2*e + 24*a^2*b*f)*x)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 15.2014, size = 354, normalized size = 3.11 \begin{align*} \begin{cases} - \frac{\frac{2 a c}{\sqrt{a + b x}} + \frac{2 a d \left (- \frac{a}{\sqrt{a + b x}} - \sqrt{a + b x}\right )}{b} + \frac{2 a e \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b^{2}} + \frac{2 a f \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{3}} + 2 c \left (- \frac{a}{\sqrt{a + b x}} - \sqrt{a + b x}\right ) + \frac{2 d \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b} + \frac{2 e \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{2}} + \frac{2 f \left (\frac{a^{4}}{\sqrt{a + b x}} + 4 a^{3} \sqrt{a + b x} - 2 a^{2} \left (a + b x\right )^{\frac{3}{2}} + \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{3}}}{b} & \text{for}\: b \neq 0 \\\frac{c x + \frac{d x^{2}}{2} + \frac{e x^{3}}{3} + \frac{f x^{4}}{4}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**3+e*x**2+d*x+c)/(b*x+a)**(1/2),x)

[Out]

Piecewise((-(2*a*c/sqrt(a + b*x) + 2*a*d*(-a/sqrt(a + b*x) - sqrt(a + b*x))/b + 2*a*e*(a**2/sqrt(a + b*x) + 2*
a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b**2 + 2*a*f*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**
(3/2) - (a + b*x)**(5/2)/5)/b**3 + 2*c*(-a/sqrt(a + b*x) - sqrt(a + b*x)) + 2*d*(a**2/sqrt(a + b*x) + 2*a*sqrt
(a + b*x) - (a + b*x)**(3/2)/3)/b + 2*e*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a
+ b*x)**(5/2)/5)/b**2 + 2*f*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*
x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**3)/b, Ne(b, 0)), ((c*x + d*x**2/2 + e*x**3/3 + f*x**4/4)/sqrt(a), True))

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Giac [A]  time = 1.08042, size = 174, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (105 \, \sqrt{b x + a} c + \frac{35 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} d}{b} + \frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} e}{b^{2}} + \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )} f}{b^{3}}\right )}}{105 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/105*(105*sqrt(b*x + a)*c + 35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*d/b + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a
)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2 + 3*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2
- 35*sqrt(b*x + a)*a^3)*f/b^3)/b