Optimal. Leaf size=405 \[ \frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{128 g^{5/2} (c d f-a e g)^{7/2}}+\frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 g^2 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{80 g^2 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}-\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5} \]
________________________________________________________________________________________
Rubi [A] time = 0.56, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {862, 872, 874, 205} \begin {gather*} \frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 g^2 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{80 g^2 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{128 g^{5/2} (c d f-a e g)^{7/2}}-\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 862
Rule 872
Rule 874
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^6} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {(3 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx}{10 g}\\ &=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {\left (3 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{80 g^2}\\ &=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {\left (c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 g^2 (c d f-a e g)}\\ &=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {\left (3 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 g^2 (c d f-a e g)^2}\\ &=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^2 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {\left (3 c^5 d^5\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 g^2 (c d f-a e g)^3}\\ &=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^2 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {\left (3 c^5 d^5 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{128 g^2 (c d f-a e g)^3}\\ &=-\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^2 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{128 g^{5/2} (c d f-a e g)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 79, normalized size = 0.20 \begin {gather*} \frac {2 c^5 d^5 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left (\frac {5}{2},6;\frac {7}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{5 (d+e x)^{5/2} (c d f-a e g)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.34, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 3204, normalized size = 7.91
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 955, normalized size = 2.36 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{5} d^{5} g^{5} x^{5} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+75 c^{5} d^{5} f \,g^{4} x^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+150 c^{5} d^{5} f^{2} g^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+150 c^{5} d^{5} f^{3} g^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+75 c^{5} d^{5} f^{4} g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+15 c^{5} d^{5} f^{5} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{4} d^{4} g^{4} x^{4}+10 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,g^{4} x^{3}-70 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{4} d^{4} f \,g^{3} x^{3}-8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}+46 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e f \,g^{3} x^{2}-128 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{4} d^{4} f^{2} g^{2} x^{2}-176 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{3} g^{4} x +512 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -466 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,f^{2} g^{2} x +70 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{4} d^{4} f^{3} g x -128 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{4} e^{4} g^{4}+336 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{3} f \,g^{3}-248 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}+10 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,f^{3} g +15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{4} d^{4} f^{4}\right )}{640 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{5} \left (a e g -c d f \right ) \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d x +a e}\, g^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (f+g\,x\right )}^6\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________