Optimal. Leaf size=313 \[ -\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{8 c^{5/2} d^{5/2} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e}{c d}-\frac {f}{g}\right )}{12 \sqrt {d+e x}} \]
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Rubi [A] time = 0.52, antiderivative size = 313, 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 = {864, 870, 891, 63, 217, 206} \[ -\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{8 c^{5/2} d^{5/2} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e}{c d}-\frac {f}{g}\right )}{12 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 217
Rule 864
Rule 870
Rule 891
Rubi steps
\begin {align*} \int \frac {(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx &=\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 g}\\ &=\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g)^2 \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 c d g}\\ &=-\frac {(c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g)^3 \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}{16 c^2 d^2 g}\\ &=-\frac {(c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {\left ((c d f-a e g)^3 \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}{16 c^2 d^2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {(c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {\left ((c d f-a e g)^3 \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 )}{8 c^3 d^3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {(c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {\left ((c d f-a e g)^3 \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 )}{8 c^3 d^3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {(c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g)^3 \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 )}{8 c^{5/2} d^{5/2} g^{3/2} \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 [A] time = 0.84, size = 255, normalized size = 0.81 \[ \frac {\sqrt {c d} \sqrt {d+e x} \left (-\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {c d} (f+g x) (a e+c d x) \left (3 a^2 e^2 g^2-2 a c d e g (4 f+g x)-c^2 d^2 \left (3 f^2+14 f g x+8 g^2 x^2\right )\right )-3 \sqrt {a e+c d x} (c d f-a e g)^{7/2} \sqrt {\frac {c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d f-a e g}}\right )\right )}{24 c^{7/2} d^{7/2} g^{3/2} \sqrt {f+g x} \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.27, size = 847, normalized size = 2.71 \[ \left [\frac {4 \, {\left (8 \, c^{3} d^{3} g^{3} x^{2} + 3 \, c^{3} d^{3} f^{2} g + 8 \, a c^{2} d^{2} e f g^{2} - 3 \, a^{2} c d e^{2} g^{3} + 2 \, {\left (7 \, c^{3} d^{3} f g^{2} + a c^{2} d^{2} e g^{3}\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} - 3 \, {\left (c^{3} d^{4} f^{3} - 3 \, a c^{2} d^{3} e f^{2} g + 3 \, a^{2} c d^{2} e^{2} f g^{2} - a^{3} d e^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g + 3 \, a^{2} c d e^{3} f g^{2} - a^{3} e^{4} g^{3}\right )} x\right )} \sqrt {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 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {c d g} \sqrt {e x + d} \sqrt {g x + f} + 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 )}{96 \, {\left (c^{3} d^{3} e g^{2} x + c^{3} d^{4} g^{2}\right )}}, \frac {2 \, {\left (8 \, c^{3} d^{3} g^{3} x^{2} + 3 \, c^{3} d^{3} f^{2} g + 8 \, a c^{2} d^{2} e f g^{2} - 3 \, a^{2} c d e^{2} g^{3} + 2 \, {\left (7 \, c^{3} d^{3} f g^{2} + a c^{2} d^{2} e g^{3}\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} + 3 \, {\left (c^{3} d^{4} f^{3} - 3 \, a c^{2} d^{3} e f^{2} g + 3 \, a^{2} c d^{2} e^{2} f g^{2} - a^{3} d e^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g + 3 \, a^{2} c d e^{3} f g^{2} - a^{3} e^{4} g^{3}\right )} x\right )} \sqrt {-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 {-c d g} \sqrt {e x + d} \sqrt {g x + f}}{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 )}{48 \, {\left (c^{3} d^{3} e g^{2} x + c^{3} d^{4} g^{2}\right )}}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 602, normalized size = 1.92 \[ \frac {\sqrt {g x +f}\, \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (3 a^{3} e^{3} g^{3} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-9 a^{2} c d \,e^{2} f \,g^{2} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+9 a \,c^{2} d^{2} e \,f^{2} g \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-3 c^{3} d^{3} f^{3} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+16 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{2} d^{2} g^{2} x^{2}+4 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a c d e \,g^{2} x +28 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{2} d^{2} f g x -6 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{2} e^{2} g^{2}+16 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a c d e f g +6 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{2} d^{2} f^{2}\right )}{48 \sqrt {e x +d}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c^{2} d^{2} g} \]
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 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {e x + d}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^{3/2}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\sqrt {d+e\,x}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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