Optimal. Leaf size=294 \[ \frac {15 \sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {15 c d \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}} \]
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Rubi [A] time = 0.43, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {862, 864, 891, 63, 217, 206} \begin {gather*} \frac {5 c d \sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {15 c d \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^3 \sqrt {d+e x}}+\frac {15 \sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 217
Rule 862
Rule 864
Rule 891
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 )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx &=-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {(5 c d) \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} \sqrt {f+g x}} \, dx}{g}\\ &=\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}-\frac {(15 c d (c d f-a e g)) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{4 g^2}\\ &=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 c d (c d f-a e g)^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g^3}\\ &=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 c d (c d f-a e g)^2 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}} \, dx}{8 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 (c d f-a e g)^2 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {a e g}{c d}+\frac {g x^2}{c d}}} \, dx,x,\sqrt {a e+c d x}\right )}{4 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 (c d f-a e g)^2 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{c d}} \, dx,x,\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}\right )}{4 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {15 \sqrt {c} \sqrt {d} (c d f-a e g)^2 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 112, normalized size = 0.38 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{7 c d \sqrt {d+e x} (f+g x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.06, size = 255, normalized size = 0.87 \begin {gather*} \frac {g^{5/2} \left (\frac {(d g+e g x) (a e g+c d g x)}{g^2}\right )^{5/2} \left (\frac {\sqrt {a e g+c d (f+g x)-c d f} \left (-8 a^2 e^2 g^2+9 a c d e g (f+g x)+16 a c d e f g-8 c^2 d^2 f^2+2 c^2 d^2 (f+g x)^2-9 c^2 d^2 f (f+g x)\right )}{4 g^{7/2} \sqrt {f+g x}}-\frac {15 \sqrt {c d} \left (a^2 e^2 g^2-2 a c d e f g+c^2 d^2 f^2\right ) \log \left (\sqrt {a e g+c d (f+g x)-c d f}-\sqrt {c d} \sqrt {f+g x}\right )}{4 g^{7/2}}\right )}{(d+e x)^{5/2} (a e g+c d g x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 915, normalized size = 3.11 \begin {gather*} \left [\frac {4 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 25 \, a c d e f g - 8 \, a^{2} e^{2} g^{2} - {\left (5 \, c^{2} d^{2} f g - 9 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} + 15 \, {\left (c^{2} d^{3} f^{3} - 2 \, a c d^{2} e f^{2} g + a^{2} d e^{2} f g^{2} + {\left (c^{2} d^{2} e f^{2} g - 2 \, a c d e^{2} f g^{2} + a^{2} e^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} + a^{2} d e^{2} g^{3} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{2}\right )} x\right )} \sqrt {\frac {c d}{g}} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \, {\left (2 \, c d g^{2} x + c d f g + a e g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} \sqrt {\frac {c d}{g}} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{16 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}, \frac {2 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 25 \, a c d e f g - 8 \, a^{2} e^{2} g^{2} - {\left (5 \, c^{2} d^{2} f g - 9 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} - 15 \, {\left (c^{2} d^{3} f^{3} - 2 \, a c d^{2} e f^{2} g + a^{2} d e^{2} f g^{2} + {\left (c^{2} d^{2} e f^{2} g - 2 \, a c d e^{2} f g^{2} + a^{2} e^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} + a^{2} d e^{2} g^{3} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{2}\right )} x\right )} \sqrt {-\frac {c d}{g}} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} \sqrt {-\frac {c d}{g}} g}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{8 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [B] time = 0.03, size = 635, normalized size = 2.16 \begin {gather*} \frac {\left (15 a^{2} c d \,e^{2} g^{3} x \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-30 a \,c^{2} d^{2} e f \,g^{2} x \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+15 c^{3} d^{3} f^{2} g x \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+15 a^{2} c d \,e^{2} f \,g^{2} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-30 a \,c^{2} d^{2} e \,f^{2} g \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+15 c^{3} d^{3} f^{3} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+4 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} g^{2} x^{2}+18 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a c d e \,g^{2} x -10 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{2} d^{2} f g x -16 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+50 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a c d e f g -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{8 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, \sqrt {g x +f}\, \sqrt {e x +d}\, g^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )}^{5/2}}{{\left (f+g\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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