Optimal. Leaf size=239 \[ -\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 g^2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e} \]
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Rubi [A]
time = 0.18, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {880, 808, 662}
\begin {gather*} -\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^3 d^3 e}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 808
Rule 880
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} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(2 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}{c d}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 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}{c^2 d^2}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e}-\frac {\left (8 g^2 \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^3 d^3 e}\\ &=-\frac {2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {4 g \sqrt {d+e x} (f+g x)^2}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 g^2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 e}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 131, normalized size = 0.55 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-6 a c^2 d^2 e g \left (f^2-6 f g x+g^2 x^2\right )+c^3 d^3 \left (-f^3-9 f^2 g x+9 f g^2 x^2+g^3 x^3\right )\right )}{3 c^4 d^4 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 179, normalized size = 0.75
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-g^{3} x^{3} c^{3} d^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} c^{4} d^{4}}\) | \(179\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-g^{3} x^{3} c^{3} d^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f \,g^{2} x^{2}+24 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x +9 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+6 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{4} d^{4} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 227, normalized size = 0.95 \begin {gather*} -\frac {2 \, {\left (3 \, c d x + 2 \, a e\right )} f^{2} g}{{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + 12 \, a c d x e + 8 \, a^{2} e^{2}\right )} f g^{2}}{{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt {c d x + a e}} + \frac {2 \, {\left (c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{2} x^{2} e - 24 \, a^{2} c d x e^{2} - 16 \, a^{3} e^{3}\right )} g^{3}}{3 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )} \sqrt {c d x + a e}} - \frac {2 \, f^{3}}{3 \, {\left (c^{2} d^{2} x + a c d e\right )} \sqrt {c d x + a e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.85, size = 247, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} g^{3} x^{3} + 9 \, c^{3} d^{3} f g^{2} x^{2} - 9 \, c^{3} d^{3} f^{2} g x - c^{3} d^{3} f^{3} - 16 \, a^{3} g^{3} e^{3} - 24 \, {\left (a^{2} c d g^{3} x - a^{2} c d f g^{2}\right )} e^{2} - 6 \, {\left (a c^{2} d^{2} g^{3} x^{2} - 6 \, a c^{2} d^{2} f g^{2} x + a c^{2} d^{2} f^{2} 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^{6} d^{7} x^{2} + a^{2} c^{4} d^{4} x e^{3} + {\left (2 \, a c^{5} d^{5} x^{2} + a^{2} c^{4} d^{5}\right )} e^{2} + {\left (c^{6} d^{6} x^{3} + 2 \, a c^{5} d^{6} x\right )} e\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 509 vs.
\(2 (221) = 442\).
time = 1.37, size = 509, normalized size = 2.13 \begin {gather*} -\frac {2 \, {\left (c^{3} d^{6} g^{3} - 9 \, c^{3} d^{5} f g^{2} e - 9 \, c^{3} d^{4} f^{2} g e^{2} + 6 \, a c^{2} d^{4} g^{3} e^{2} + c^{3} d^{3} f^{3} e^{3} + 36 \, a c^{2} d^{3} f g^{2} e^{3} + 6 \, a c^{2} d^{2} f^{2} g e^{4} - 24 \, a^{2} c d^{2} g^{3} e^{4} - 24 \, a^{2} c d f g^{2} e^{5} + 16 \, a^{3} g^{3} e^{6}\right )}}{3 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{5} d^{6} e - \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{4} e^{3}\right )}} - \frac {2 \, {\left (c^{3} d^{3} f^{3} e^{3} - 3 \, a c^{2} d^{2} f^{2} g e^{4} + 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} f^{2} g e + 3 \, a^{2} c d f g^{2} e^{5} - 18 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a c d f g^{2} e^{2} - a^{3} g^{3} e^{6} + 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{2} g^{3} e^{3}\right )}}{3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{4}} + \frac {2 \, {\left (9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{9} d^{9} f g^{2} e^{8} - 9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{8} d^{8} g^{3} e^{9} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{8} d^{8} g^{3} e^{6}\right )} e^{\left (-9\right )}}{3 \, c^{12} d^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.77, size = 278, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (\frac {32\,a^3\,e^3\,g^3}{3}-16\,a^2\,c\,d\,e^2\,f\,g^2+4\,a\,c^2\,d^2\,e\,f^2\,g+\frac {2\,c^3\,d^3\,f^3}{3}\right )}{c^6\,d^6\,e}-\frac {2\,g^3\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,d^3\,e}+\frac {g^2\,x^2\,\left (4\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c^4\,d^4\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^2\,g^2-12\,a\,c\,d\,e\,f\,g+3\,c^2\,d^2\,f^2\right )}{c^5\,d^5\,e}\right )}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c^6\,d^7+2\,a\,c^5\,d^5\,e^2\right )}{c^6\,d^6\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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