Optimal. Leaf size=342 \[ -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {35 c^2 d^2 g^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 (c d f-a e g)^{9/2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {35 c^2 d^2 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 )}{4 (c d f-a e g)^{9/2}}+\frac {35 c d g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt {d+e x} (f+g x) (c d f-a e g)^4}+\frac {35 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^3}+\frac {14 g \sqrt {d+e x}}{3 (f+g x)^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}-\frac {2 (d+e x)^{3/2}}{3 (f+g x)^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)} \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)^3 \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)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(7 g) \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \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)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 g^2\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}{3 (c d f-a e g)^2}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (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}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {\left (35 c d 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}{4 (c d f-a e g)^3}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (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}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {\left (35 c^2 d^2 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}{8 (c d f-a e g)^4}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (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}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {\left (35 c^2 d^2 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 )}{4 (c d f-a e g)^4}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (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}}+\frac {14 g \sqrt {d+e x}}{3 (c d f-a e g)^2 (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^2}+\frac {35 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)}+\frac {35 c^2 d^2 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 )}{4 (c d f-a e g)^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 240, normalized size = 0.70 \begin {gather*} \frac {c^2 d^2 \sqrt {d+e x} \left (\frac {-6 a^3 e^3 g^3+3 a^2 c d e^2 g^2 (13 f+7 g x)+2 a c^2 d^2 e g \left (40 f^2+119 f g x+70 g^2 x^2\right )+c^3 d^3 \left (-8 f^3+56 f^2 g x+175 f g^2 x^2+105 g^3 x^3\right )}{c^2 d^2 (c d f-a e g)^4 (a e+c d x) (f+g x)^2}+\frac {105 g^{3/2} \sqrt {a e+c d x} \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)^{9/2}}\right )}{12 \sqrt {(a e+c d x) (d+e x)}} \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. \(659\) vs.
\(2(304)=608\).
time = 0.14, size = 660, normalized size = 1.93
method | result | size |
default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} g^{4} x^{3} \sqrt {c d x +a e}+105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,g^{4} x^{2} \sqrt {c d x +a e}+210 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{3} x^{2} \sqrt {c d x +a e}+210 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e f \,g^{3} x \sqrt {c d x +a e}+105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g^{2} x \sqrt {c d x +a e}-105 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} g^{3} x^{3}+105 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,f^{2} g^{2} \sqrt {c d x +a e}-140 \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,g^{3} x^{2}-175 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f \,g^{2} x^{2}-21 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} g^{3} x -238 \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e f \,g^{2} x -56 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{2} g x +6 \sqrt {\left (a e g -c d f \right ) g}\, a^{3} e^{3} g^{3}-39 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} f \,g^{2}-80 \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,f^{2} g +8 \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{3}\right )}{12 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{4} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\) | \(660\) |
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. 1501 vs.
\(2 (320) = 640\).
time = 6.27, size = 3043, normalized size = 8.90 \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 (d+e\,x\right )}^{5/2}}{{\left (f+g\,x\right )}^3\,{\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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