Integrand size = 25, antiderivative size = 114 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt {a+b x}}{b^4}+\frac {2 \left (b^2 d-2 a b e+3 a^2 f\right ) (a+b x)^{3/2}}{3 b^4}+\frac {2 (b e-3 a f) (a+b x)^{5/2}}{5 b^4}+\frac {2 f (a+b x)^{7/2}}{7 b^4} \] Output:
2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*(b*x+a)^(1/2)/b^4+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
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.72 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (-48 a^3 f+8 a^2 b (7 e+3 f x)-2 a b^2 (35 d+x (14 e+9 f x))+b^3 (105 c+x (35 d+3 x (7 e+5 f x)))\right )}{105 b^4} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/Sqrt[a + b*x],x]
Output:
(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*(1 4*e + 9*f*x)) + b^3*(105*c + x*(35*d + 3*x*(7*e + 5*f*x)))))/(105*b^4)
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2389, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {\sqrt {a+b x} \left (3 a^2 f-2 a b e+b^2 d\right )}{b^3}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{b^3 \sqrt {a+b x}}+\frac {(a+b x)^{3/2} (b e-3 a f)}{b^3}+\frac {f (a+b x)^{5/2}}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 (a+b x)^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac {2 \sqrt {a+b x} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac {2 (a+b x)^{5/2} (b e-3 a f)}{5 b^4}+\frac {2 f (a+b x)^{7/2}}{7 b^4}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/Sqrt[a + b*x],x]
Output:
(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)
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])
Time = 0.36 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {32 \left (\frac {\left (-5 f \,x^{3}-7 e \,x^{2}-\frac {35}{3} x d -35 c \right ) b^{3}}{16}+\frac {35 a \left (\frac {9}{35} f \,x^{2}+\frac {2}{5} e x +d \right ) b^{2}}{24}-\frac {7 a^{2} \left (\frac {3 f x}{7}+e \right ) b}{6}+a^{3} f \right ) \sqrt {b x +a}}{35 b^{4}}\) | \(74\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-15 f \,x^{3} b^{3}+18 a \,b^{2} f \,x^{2}-21 b^{3} e \,x^{2}-24 a^{2} b f x +28 a \,b^{2} e x -35 b^{3} d x +48 a^{3} f -56 a^{2} b e +70 a \,b^{2} d -105 b^{3} c \right )}{105 b^{4}}\) | \(91\) |
trager | \(-\frac {2 \sqrt {b x +a}\, \left (-15 f \,x^{3} b^{3}+18 a \,b^{2} f \,x^{2}-21 b^{3} e \,x^{2}-24 a^{2} b f x +28 a \,b^{2} e x -35 b^{3} d x +48 a^{3} f -56 a^{2} b e +70 a \,b^{2} d -105 b^{3} c \right )}{105 b^{4}}\) | \(91\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-15 f \,x^{3} b^{3}+18 a \,b^{2} f \,x^{2}-21 b^{3} e \,x^{2}-24 a^{2} b f x +28 a \,b^{2} e x -35 b^{3} d x +48 a^{3} f -56 a^{2} b e +70 a \,b^{2} d -105 b^{3} c \right )}{105 b^{4}}\) | \(91\) |
orering | \(-\frac {2 \sqrt {b x +a}\, \left (-15 f \,x^{3} b^{3}+18 a \,b^{2} f \,x^{2}-21 b^{3} e \,x^{2}-24 a^{2} b f x +28 a \,b^{2} e x -35 b^{3} d x +48 a^{3} f -56 a^{2} b e +70 a \,b^{2} d -105 b^{3} c \right )}{105 b^{4}}\) | \(91\) |
derivativedivides | \(\frac {\frac {2 f \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {6 a f \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 b e \left (b x +a \right )^{\frac {5}{2}}}{5}+2 a^{2} f \left (b x +a \right )^{\frac {3}{2}}-\frac {4 a b e \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} d \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a^{3} f \sqrt {b x +a}+2 a^{2} b e \sqrt {b x +a}-2 a \,b^{2} d \sqrt {b x +a}+2 b^{3} c \sqrt {b x +a}}{b^{4}}\) | \(128\) |
default | \(\frac {\frac {2 f \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {6 a f \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 b e \left (b x +a \right )^{\frac {5}{2}}}{5}+2 a^{2} f \left (b x +a \right )^{\frac {3}{2}}-\frac {4 a b e \left (b x +a \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} d \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a^{3} f \sqrt {b x +a}+2 a^{2} b e \sqrt {b x +a}-2 a \,b^{2} d \sqrt {b x +a}+2 b^{3} c \sqrt {b x +a}}{b^{4}}\) | \(128\) |
Input:
int((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-32/35*(1/16*(-5*f*x^3-7*e*x^2-35/3*x*d-35*c)*b^3+35/24*a*(9/35*f*x^2+2/5* e*x+d)*b^2-7/6*a^2*(3/7*f*x+e)*b+a^3*f)*(b*x+a)^(1/2)/b^4
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.79 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (15 \, b^{3} f x^{3} + 105 \, b^{3} c - 70 \, a b^{2} d + 56 \, a^{2} b e - 48 \, a^{3} f + 3 \, {\left (7 \, b^{3} e - 6 \, a b^{2} f\right )} x^{2} + {\left (35 \, b^{3} d - 28 \, a b^{2} e + 24 \, a^{2} b f\right )} x\right )} \sqrt {b x + a}}{105 \, b^{4}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
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
Time = 0.58 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 c \sqrt {a + b x} + \frac {2 d \left (- a \sqrt {a + b x} + \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b} + \frac {2 e \left (a^{2} \sqrt {a + b x} - \frac {2 a \left (a + b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a + b x\right )^{\frac {5}{2}}}{5}\right )}{b^{2}} + \frac {2 f \left (- a^{3} \sqrt {a + b x} + a^{2} \left (a + b x\right )^{\frac {3}{2}} - \frac {3 a \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {\left (a + b x\right )^{\frac {7}{2}}}{7}\right )}{b^{3}}}{b} & \text {for}\: b \neq 0 \\\frac {c x + \frac {d x^{2}}{2} + \frac {e x^{3}}{3} + \frac {f x^{4}}{4}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/(b*x+a)**(1/2),x)
Output:
Piecewise(((2*c*sqrt(a + b*x) + 2*d*(-a*sqrt(a + b*x) + (a + b*x)**(3/2)/3 )/b + 2*e*(a**2*sqrt(a + b*x) - 2*a*(a + b*x)**(3/2)/3 + (a + b*x)**(5/2)/ 5)/b**2 + 2*f*(-a**3*sqrt(a + b*x) + a**2*(a + b*x)**(3/2) - 3*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))
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (105 \, \sqrt {b x + a} c + \frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} f}{b^{3}}\right )}}{105 \, b} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
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
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (105 \, \sqrt {b x + a} c + \frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} f}{b^{3}}\right )}}{105 \, b} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x, algorithm="giac")
Output:
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
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {{\left (a+b\,x\right )}^{3/2}\,\left (6\,f\,a^2-4\,e\,a\,b+2\,d\,b^2\right )}{3\,b^4}-\frac {\left (6\,a\,f-2\,b\,e\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}+\frac {\sqrt {a+b\,x}\,\left (-2\,f\,a^3+2\,e\,a^2\,b-2\,d\,a\,b^2+2\,c\,b^3\right )}{b^4}+\frac {2\,f\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4} \] Input:
int((c + d*x + e*x^2 + f*x^3)/(a + b*x)^(1/2),x)
Output:
((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)
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x+e x^2+f x^3}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {b x +a}\, \left (15 b^{3} f \,x^{3}-18 a \,b^{2} f \,x^{2}+21 b^{3} e \,x^{2}+24 a^{2} b f x -28 a \,b^{2} e x +35 b^{3} d x -48 a^{3} f +56 a^{2} b e -70 a \,b^{2} d +105 b^{3} c \right )}{105 b^{4}} \] Input:
int((f*x^3+e*x^2+d*x+c)/(b*x+a)^(1/2),x)
Output:
(2*sqrt(a + b*x)*( - 48*a**3*f + 56*a**2*b*e + 24*a**2*b*f*x - 70*a*b**2*d - 28*a*b**2*e*x - 18*a*b**2*f*x**2 + 105*b**3*c + 35*b**3*d*x + 21*b**3*e *x**2 + 15*b**3*f*x**3))/(105*b**4)