Optimal. Leaf size=104 \[ -\frac {(d+e x)^{3/2} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left (1,n+\frac {1}{2};n+2;\frac {c d (f+g x)}{c d f-a e g}\right )}{(n+1) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 110, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {891, 70, 69} \[ -\frac {2 \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac {1}{2},-n;\frac {1}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 891
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^n}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {\left (\sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {(f+g x)^n}{(a e+c d x)^{3/2}} \, dx}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {\left (\sqrt {a e+c d x} \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n}\right ) \int \frac {\left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^n}{(a e+c d x)^{3/2}} \, dx}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {2 \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac {1}{2},-n;\frac {1}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 98, normalized size = 0.94 \[ -\frac {2 \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac {1}{2},-n;\frac {1}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{c d \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} {\left (g x + f\right )}^{n}}{c^{2} d^{2} e x^{3} + a^{2} d e^{2} + {\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{2} + {\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right )^{n}}{\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________