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

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

[Out]

2/3*(3*a^2*f-2*a*b*e+b^2*d)*(b*x+a)^(3/2)/b^4+2/5*(-3*a*f+b*e)*(b*x+a)^(5/2)/b^4+2/7*f*(b*x+a)^(7/2)/b^4+2*(-a
^3*f+a^2*b*e-a*b^2*d+b^3*c)*(b*x+a)^(1/2)/b^4

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Rubi [A]  time = 0.07, antiderivative size = 114, 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} \[ \frac {2 \sqrt {a+b x} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}+\frac {2 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac {2 (a+b x)^{5/2} (b e-3 a f)}{5 b^4}+\frac {2 f (a+b x)^{7/2}}{7 b^4} \]

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*}

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Mathematica [A]  time = 0.18, size = 82, normalized size = 0.72 \[ \frac {2 \sqrt {a+b x} \left (-48 a^3 f+8 a^2 b (7 e+3 f x)-2 a b^2 (35 d+x (14 e+9 f x))+b^3 (105 c+x (35 d+3 x (7 e+5 f x)))\right )}{105 b^4} \]

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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fricas [A]  time = 0.60, size = 90, normalized size = 0.79 \[ \frac {2 \, {\left (15 \, b^{3} f x^{3} + 105 \, b^{3} c - 70 \, a b^{2} d + 56 \, a^{2} b e - 48 \, a^{3} f + 3 \, {\left (7 \, b^{3} e - 6 \, a b^{2} f\right )} x^{2} + {\left (35 \, b^{3} d - 28 \, a b^{2} e + 24 \, a^{2} b f\right )} x\right )} \sqrt {b x + a}}{105 \, b^{4}} \]

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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giac [A]  time = 0.17, size = 129, normalized size = 1.13 \[ \frac {2 \, {\left (105 \, \sqrt {b x + a} c + \frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} f}{b^{3}}\right )}}{105 \, b} \]

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

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maple [A]  time = 0.04, size = 91, normalized size = 0.80 \[ -\frac {2 \sqrt {b x +a}\, \left (-15 f \,x^{3} b^{3}+18 a \,b^{2} f \,x^{2}-21 b^{3} e \,x^{2}-24 a^{2} b f x +28 a \,b^{2} e x -35 b^{3} d x +48 a^{3} f -56 a^{2} b e +70 a \,b^{2} d -105 b^{3} c \right )}{105 b^{4}} \]

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 = 0.83, size = 128, normalized size = 1.12 \[ \frac {2 \, {\left (105 \, \sqrt {b x + a} c + \frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} f}{b^{3}}\right )}}{105 \, b} \]

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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mupad [B]  time = 4.81, size = 103, normalized size = 0.90 \[ \frac {{\left (a+b\,x\right )}^{3/2}\,\left (6\,f\,a^2-4\,e\,a\,b+2\,d\,b^2\right )}{3\,b^4}-\frac {\left (6\,a\,f-2\,b\,e\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}+\frac {\sqrt {a+b\,x}\,\left (-2\,f\,a^3+2\,e\,a^2\,b-2\,d\,a\,b^2+2\,c\,b^3\right )}{b^4}+\frac {2\,f\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a + b*x)^(3/2)*(2*b^2*d + 6*a^2*f - 4*a*b*e))/(3*b^4) - ((6*a*f - 2*b*e)*(a + b*x)^(5/2))/(5*b^4) + ((a + b*
x)^(1/2)*(2*b^3*c - 2*a^3*f - 2*a*b^2*d + 2*a^2*b*e))/b^4 + (2*f*(a + b*x)^(7/2))/(7*b^4)

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sympy [A]  time = 45.83, size = 354, normalized size = 3.11 \[ \begin {cases} \frac {- \frac {2 a c}{\sqrt {a + b x}} - \frac {2 a d \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right )}{b} - \frac {2 a e \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b^{2}} - \frac {2 a f \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{3}} - 2 c \left (- \frac {a}{\sqrt {a + b x}} - \sqrt {a + b x}\right ) - \frac {2 d \left (\frac {a^{2}}{\sqrt {a + b x}} + 2 a \sqrt {a + b x} - \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b} - \frac {2 e \left (- \frac {a^{3}}{\sqrt {a + b x}} - 3 a^{2} \sqrt {a + b x} + a \left (a + b x\right )^{\frac {3}{2}} - \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}} - \frac {2 f \left (\frac {a^{4}}{\sqrt {a + b x}} + 4 a^{3} \sqrt {a + b x} - 2 a^{2} \left (a + b x\right )^{\frac {3}{2}} + \frac {4 a \left (a + b x\right )^{\frac {5}{2}}}{5} - \frac {\left (a + b x\right )^{\frac {7}{2}}}{7}\right )}{b^{3}}}{b} & \text {for}\: b \neq 0 \\\frac {c x + \frac {d x^{2}}{2} + \frac {e x^{3}}{3} + \frac {f x^{4}}{4}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

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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