Optimal. Leaf size=405 \[ -\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{40 g^2 \sqrt {d+e x} (f+g x)^4}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{80 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^2 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{128 g^{5/2} (c d f-a e g)^{7/2}} \]
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Rubi [A]
time = 0.41, 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 = {876, 886, 888,
211} \begin {gather*} \frac {3 c^5 d^5 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 876
Rule 886
Rule 888
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 ) \text {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*}
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Mathematica [A]
time = 2.39, size = 302, normalized size = 0.75 \begin {gather*} \frac {c^5 d^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {g} \left (128 a^4 e^4 g^4+16 a^3 c d e^3 g^3 (-21 f+11 g x)+8 a^2 c^2 d^2 e^2 g^2 \left (31 f^2-64 f g x+g^2 x^2\right )-2 a c^3 d^3 e g \left (5 f^3-233 f^2 g x+23 f g^2 x^2+5 g^3 x^3\right )+c^4 d^4 \left (-15 f^4-70 f^3 g x+128 f^2 g^2 x^2+70 f g^3 x^3+15 g^4 x^4\right )\right )}{c^5 d^5 (c d f-a e g)^3 (a e+c d x) (f+g x)^5}+\frac {15 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{7/2} (a e+c d x)^{3/2}}\right )}{640 g^{5/2} (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(944\) vs.
\(2(361)=722\).
time = 0.13, size = 945, normalized size = 2.33
method | result | size |
default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} g^{5} x^{5}+75 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f \,g^{4} x^{4}+150 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{2} g^{3} x^{3}+150 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{3} g^{2} x^{2}-15 c^{4} d^{4} g^{4} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+75 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{4} g x +10 a \,c^{3} d^{3} e \,g^{4} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-70 c^{4} d^{4} f \,g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{5}-8 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+46 a \,c^{3} d^{3} e f \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-128 c^{4} d^{4} f^{2} g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-176 a^{3} c d \,e^{3} g^{4} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+512 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-466 a \,c^{3} d^{3} e \,f^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+70 c^{4} d^{4} f^{3} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-128 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{4} e^{4} g^{4}+336 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} c d \,e^{3} f \,g^{3}-248 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}+10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{3} d^{3} e \,f^{3} g +15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, 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} g^{2} \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}}\) | \(945\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1653 vs.
\(2 (378) = 756\).
time = 7.49, size = 3345, normalized size = 8.26 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F(-1)] Timed out
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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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.
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