Optimal. Leaf size=268 \[ -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {10 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c d g^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{7/2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {882, 886, 888,
211} \begin {gather*} \frac {5 c d g^{3/2} \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 )}{(c d f-a e g)^{7/2}}+\frac {5 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {10 g \sqrt {d+e x}}{3 (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}-\frac {2 (d+e x)^{3/2}}{3 (f+g x) \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 882
Rule 886
Rule 888
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(5 g) \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 (c d f-a e g)}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {10 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 g^2\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}{(c d f-a e g)^2}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {10 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c d g^2\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}{2 (c d f-a e g)^3}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {10 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c d e^2 g^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 )}{(c d f-a e g)^3}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {10 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c d g^{3/2} \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 )}{(c d f-a e g)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 180, normalized size = 0.67 \begin {gather*} \frac {(d+e x)^{3/2} \left (\sqrt {c d f-a e g} \left (3 a^2 e^2 g^2+2 a c d e g (7 f+10 g x)+c^2 d^2 \left (-2 f^2+10 f g x+15 g^2 x^2\right )\right )+15 c d g^{3/2} (a e+c d x)^{3/2} (f+g x) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 (c d f-a e g)^{7/2} ((a e+c d x) (d+e x))^{3/2} (f+g x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 414, normalized size = 1.54
method | result | size |
default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} g^{3} x^{2}+15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d e \,g^{3} x \sqrt {c d x +a e}+15 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f \,g^{2} x +15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d e f \,g^{2} \sqrt {c d x +a e}-15 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}-20 \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x -10 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x -3 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}-14 \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g +2 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{3} \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}}\) | \(414\) |
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. 964 vs.
\(2 (253) = 506\).
time = 3.62, size = 1969, normalized size = 7.35 \begin {gather*} \left [-\frac {15 \, {\left (c^{3} d^{4} g^{2} x^{3} + c^{3} d^{4} f g x^{2} + {\left (a^{2} c d g^{2} x^{2} + a^{2} c d f g x\right )} e^{3} + {\left (2 \, a c^{2} d^{2} g^{2} x^{3} + 2 \, a c^{2} d^{2} f g x^{2} + a^{2} c d^{2} g^{2} x + a^{2} c d^{2} f g\right )} e^{2} + {\left (c^{3} d^{3} g^{2} x^{4} + c^{3} d^{3} f g x^{3} + 2 \, a c^{2} d^{3} g^{2} x^{2} + 2 \, a c^{2} d^{3} f g x\right )} e\right )} \sqrt {-\frac {g}{c d f - a g e}} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} - 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d f - a g e\right )} \sqrt {x e + d} \sqrt {-\frac {g}{c d f - a g e}} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) - 2 \, {\left (15 \, c^{2} d^{2} g^{2} x^{2} + 10 \, c^{2} d^{2} f g x - 2 \, c^{2} d^{2} f^{2} + 3 \, a^{2} g^{2} e^{2} + 2 \, {\left (10 \, a c d g^{2} x + 7 \, a c d f g\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{6 \, {\left (c^{5} d^{6} f^{3} g x^{3} + c^{5} d^{6} f^{4} x^{2} - {\left (a^{5} g^{4} x^{2} + a^{5} f g^{3} x\right )} e^{6} - {\left (2 \, a^{4} c d g^{4} x^{3} - a^{4} c d f g^{3} x^{2} + a^{5} d f g^{3} - {\left (3 \, a^{4} c d f^{2} g^{2} - a^{5} d g^{4}\right )} x\right )} e^{5} - {\left (a^{3} c^{2} d^{2} g^{4} x^{4} - 5 \, a^{3} c^{2} d^{2} f g^{3} x^{3} - 3 \, a^{4} c d^{2} f^{2} g^{2} - {\left (3 \, a^{3} c^{2} d^{2} f^{2} g^{2} - 2 \, a^{4} c d^{2} g^{4}\right )} x^{2} + {\left (3 \, a^{3} c^{2} d^{2} f^{3} g - a^{4} c d^{2} f g^{3}\right )} x\right )} e^{4} + {\left (3 \, a^{2} c^{3} d^{3} f g^{3} x^{4} - 3 \, a^{3} c^{2} d^{3} f^{3} g - {\left (3 \, a^{2} c^{3} d^{3} f^{2} g^{2} + a^{3} c^{2} d^{3} g^{4}\right )} x^{3} - 5 \, {\left (a^{2} c^{3} d^{3} f^{3} g - a^{3} c^{2} d^{3} f g^{3}\right )} x^{2} + {\left (a^{2} c^{3} d^{3} f^{4} + 3 \, a^{3} c^{2} d^{3} f^{2} g^{2}\right )} x\right )} e^{3} - {\left (3 \, a c^{4} d^{4} f^{2} g^{2} x^{4} + 5 \, a^{2} c^{3} d^{4} f^{3} g x - a^{2} c^{3} d^{4} f^{4} + {\left (a c^{4} d^{4} f^{3} g - 3 \, a^{2} c^{3} d^{4} f g^{3}\right )} x^{3} - {\left (2 \, a c^{4} d^{4} f^{4} - 3 \, a^{2} c^{3} d^{4} f^{2} g^{2}\right )} x^{2}\right )} e^{2} + {\left (c^{5} d^{5} f^{3} g x^{4} - a c^{4} d^{5} f^{3} g x^{2} + 2 \, a c^{4} d^{5} f^{4} x + {\left (c^{5} d^{5} f^{4} - 3 \, a c^{4} d^{5} f^{2} g^{2}\right )} x^{3}\right )} e\right )}}, \frac {15 \, {\left (c^{3} d^{4} g^{2} x^{3} + c^{3} d^{4} f g x^{2} + {\left (a^{2} c d g^{2} x^{2} + a^{2} c d f g x\right )} e^{3} + {\left (2 \, a c^{2} d^{2} g^{2} x^{3} + 2 \, a c^{2} d^{2} f g x^{2} + a^{2} c d^{2} g^{2} x + a^{2} c d^{2} f g\right )} e^{2} + {\left (c^{3} d^{3} g^{2} x^{4} + c^{3} d^{3} f g x^{3} + 2 \, a c^{2} d^{3} g^{2} x^{2} + 2 \, a c^{2} d^{3} f g x\right )} e\right )} \sqrt {\frac {g}{c d f - a g e}} \arctan \left (-\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d f - a g e\right )} \sqrt {x e + d} \sqrt {\frac {g}{c d f - a g e}}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) + {\left (15 \, c^{2} d^{2} g^{2} x^{2} + 10 \, c^{2} d^{2} f g x - 2 \, c^{2} d^{2} f^{2} + 3 \, a^{2} g^{2} e^{2} + 2 \, {\left (10 \, a c d g^{2} x + 7 \, a c d f g\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3 \, {\left (c^{5} d^{6} f^{3} g x^{3} + c^{5} d^{6} f^{4} x^{2} - {\left (a^{5} g^{4} x^{2} + a^{5} f g^{3} x\right )} e^{6} - {\left (2 \, a^{4} c d g^{4} x^{3} - a^{4} c d f g^{3} x^{2} + a^{5} d f g^{3} - {\left (3 \, a^{4} c d f^{2} g^{2} - a^{5} d g^{4}\right )} x\right )} e^{5} - {\left (a^{3} c^{2} d^{2} g^{4} x^{4} - 5 \, a^{3} c^{2} d^{2} f g^{3} x^{3} - 3 \, a^{4} c d^{2} f^{2} g^{2} - {\left (3 \, a^{3} c^{2} d^{2} f^{2} g^{2} - 2 \, a^{4} c d^{2} g^{4}\right )} x^{2} + {\left (3 \, a^{3} c^{2} d^{2} f^{3} g - a^{4} c d^{2} f g^{3}\right )} x\right )} e^{4} + {\left (3 \, a^{2} c^{3} d^{3} f g^{3} x^{4} - 3 \, a^{3} c^{2} d^{3} f^{3} g - {\left (3 \, a^{2} c^{3} d^{3} f^{2} g^{2} + a^{3} c^{2} d^{3} g^{4}\right )} x^{3} - 5 \, {\left (a^{2} c^{3} d^{3} f^{3} g - a^{3} c^{2} d^{3} f g^{3}\right )} x^{2} + {\left (a^{2} c^{3} d^{3} f^{4} + 3 \, a^{3} c^{2} d^{3} f^{2} g^{2}\right )} x\right )} e^{3} - {\left (3 \, a c^{4} d^{4} f^{2} g^{2} x^{4} + 5 \, a^{2} c^{3} d^{4} f^{3} g x - a^{2} c^{3} d^{4} f^{4} + {\left (a c^{4} d^{4} f^{3} g - 3 \, a^{2} c^{3} d^{4} f g^{3}\right )} x^{3} - {\left (2 \, a c^{4} d^{4} f^{4} - 3 \, a^{2} c^{3} d^{4} f^{2} g^{2}\right )} x^{2}\right )} e^{2} + {\left (c^{5} d^{5} f^{3} g x^{4} - a c^{4} d^{5} f^{3} g x^{2} + 2 \, a c^{4} d^{5} f^{4} x + {\left (c^{5} d^{5} f^{4} - 3 \, a c^{4} d^{5} f^{2} g^{2}\right )} x^{3}\right )} e\right )}}\right ] \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 (d+e\,x\right )}^{5/2}}{{\left (f+g\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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