Optimal. Leaf size=448 \[ -\frac {3 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {3 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt {d+e x}}-\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]
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Rubi [A] time = 0.89, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {3 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {3 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt {d+e x}}-\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}} \]
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} \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}} \, dx &=\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \int \frac {(f+g x)^{3/2} \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}} \, dx}{2 g}\\ &=-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}+\frac {\left (3 (c d f-a e g)^2\right ) \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}{16 g^2}\\ &=\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g)^3 \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}{32 g^3}\\ &=-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^4\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}{128 c d g^3}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5\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}{256 c^2 d^2 g^3}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \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}{256 c^2 d^2 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \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 )}{128 c^3 d^3 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \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 )}{128 c^3 d^3 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {3 (c d f-a e g)^5 \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 )}{128 c^{5/2} d^{5/2} 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 [A] time = 6.01, size = 285, normalized size = 0.64 \[ \frac {\sqrt {f+g x} ((d+e x) (a e+c d x))^{7/2} \left (-\frac {15 \sqrt {c} \sqrt {d} \sqrt {c d} (c d f-a e g)^{9/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d f-a e g}}\right )}{g^{7/2} (a e+c d x)^{7/2} \sqrt {\frac {c d (f+g x)}{c d f-a e g}}}+\frac {15 c d (c d f-a e g)^4}{g^3 (a e+c d x)^3}-\frac {10 c d (c d f-a e g)^3}{g^2 (a e+c d x)^2}+\frac {8 c d (c d f-a e g)^2}{g (a e+c d x)}+48 c d (c d f-a e g)+128 c^2 d^2 (f+g x)\right )}{640 c^3 d^3 (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 11.97, size = 1331, normalized size = 2.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [B] time = 0.03, size = 1191, normalized size = 2.66 \[ \frac {\sqrt {g x +f}\, \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 a^{5} e^{5} g^{5} \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 )-75 a^{4} c d \,e^{4} f \,g^{4} \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 )+150 a^{3} c^{2} d^{2} e^{3} f^{2} 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 )-150 a^{2} c^{3} d^{3} e^{2} f^{3} 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 )+75 a \,c^{4} d^{4} e \,f^{4} 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 )-15 c^{5} d^{5} f^{5} \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 )+256 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c^{4} d^{4} g^{4} x^{4}+672 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, a \,c^{3} d^{3} e \,g^{4} x^{3}+352 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c^{4} d^{4} f \,g^{3} x^{3}+496 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}+1024 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, a \,c^{3} d^{3} e f \,g^{3} x^{2}+16 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c^{4} d^{4} f^{2} g^{2} x^{2}+20 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{3} c d \,e^{3} g^{4} x +932 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{2} c^{2} d^{2} e^{2} f \,g^{3} x +92 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a \,c^{3} d^{3} e \,f^{2} g^{2} x -20 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{4} d^{4} f^{3} g x -30 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{4} e^{4} g^{4}+140 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{3} c d \,e^{3} f \,g^{3}+256 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-140 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a \,c^{3} d^{3} e \,f^{3} g +30 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{4} d^{4} f^{4}\right )}{1280 \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^{3}} \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^{3/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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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