Optimal. Leaf size=269 \[ -\frac {16 (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 269, 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 = {884, 808, 662}
\begin {gather*} -\frac {16 \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)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac {16 g \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)^2}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (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)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 808
Rule 884
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx &=\frac {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}}{9 c d (d+e x)^{3/2}}+\frac {\left (2 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{3 c d e^2}\\ &=\frac {4 (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}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{3/2}}+\frac {\left (8 (c d f-a e g)^2\right ) \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{21 c^2 d^2}\\ &=\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (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}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{3/2}}+\frac {\left (8 (c d f-a e g)^2 \left (5 f-\frac {3 d g}{e}-\frac {2 a e g}{c d}\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{105 c^2 d^2}\\ &=\frac {16 (c d f-a e g)^2 \left (5 f-\frac {3 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 c^3 d^3 (d+e x)^{3/2}}+\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e \sqrt {d+e x}}+\frac {4 (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}}{21 c^2 d^2 (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 136, normalized size = 0.51 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (3 f+g x)-6 a c^2 d^2 e g \left (21 f^2+18 f g x+5 g^2 x^2\right )+c^3 d^3 \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 178, normalized size = 0.66
method | result | size |
default | \(-\frac {2 \left (c d x +a e \right ) \left (-35 g^{3} x^{3} c^{3} d^{3}+30 a \,c^{2} d^{2} e \,g^{3} x^{2}-135 c^{3} d^{3} f \,g^{2} x^{2}-24 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -189 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-72 a^{2} c d \,e^{2} f \,g^{2}+126 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{315 c^{4} d^{4} \sqrt {e x +d}}\) | \(178\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-35 g^{3} x^{3} c^{3} d^{3}+30 a \,c^{2} d^{2} e \,g^{3} x^{2}-135 c^{3} d^{3} f \,g^{2} x^{2}-24 a^{2} c d \,e^{2} g^{3} x +108 a \,c^{2} d^{2} e f \,g^{2} x -189 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-72 a^{2} c d \,e^{2} f \,g^{2}+126 a \,c^{2} d^{2} e \,f^{2} g -105 f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 c^{4} d^{4} \sqrt {e x +d}}\) | \(188\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 219, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f^{3}}{3 \, c d} + \frac {2 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d x e - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{2} g}{5 \, c^{2} d^{2}} + \frac {2 \, {\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} x^{2} e - 4 \, a^{2} c d x e^{2} + 8 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f g^{2}}{35 \, c^{3} d^{3}} + \frac {2 \, {\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} x^{3} e - 6 \, a^{2} c^{2} d^{2} x^{2} e^{2} + 8 \, a^{3} c d x e^{3} - 16 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} g^{3}}{315 \, c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.30, size = 263, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} d^{4} g^{3} x^{4} + 135 \, c^{4} d^{4} f g^{2} x^{3} + 189 \, c^{4} d^{4} f^{2} g x^{2} + 105 \, c^{4} d^{4} f^{3} x - 16 \, a^{4} g^{3} e^{4} + 8 \, {\left (a^{3} c d g^{3} x + 9 \, a^{3} c d f g^{2}\right )} e^{3} - 6 \, {\left (a^{2} c^{2} d^{2} g^{3} x^{2} + 6 \, a^{2} c^{2} d^{2} f g^{2} x + 21 \, a^{2} c^{2} d^{2} f^{2} g\right )} e^{2} + {\left (5 \, a c^{3} d^{3} g^{3} x^{3} + 27 \, a c^{3} d^{3} f g^{2} x^{2} + 63 \, a c^{3} d^{3} f^{2} g x + 105 \, a c^{3} d^{3} f^{3}\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}}{315 \, {\left (c^{4} d^{4} x e + c^{4} d^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{3}}{\sqrt {d + e x}}\, dx \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. 747 vs.
\(2 (253) = 506\).
time = 3.89, size = 747, normalized size = 2.78 \begin {gather*} \frac {2}{315} \, {\left (105 \, f^{3} {\left (\frac {{\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{c d} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d}\right )} e^{\left (-1\right )} + 9 \, f g^{2} {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} e^{\left (-1\right )} - g^{3} {\left (\frac {{\left (35 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} e^{2} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} e^{6} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} e^{8}\right )} e^{\left (-3\right )}}{c^{4} d^{4}} + \frac {{\left (105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} e^{\left (-7\right )}}{c^{4} d^{4}}\right )} e^{\left (-1\right )} - 63 \, f^{2} g {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} e^{\left (-2\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 242, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^3\,x^4}{9}-\frac {32\,a^4\,e^4\,g^3-144\,a^3\,c\,d\,e^3\,f\,g^2+252\,a^2\,c^2\,d^2\,e^2\,f^2\,g-210\,a\,c^3\,d^3\,e\,f^3}{315\,c^4\,d^4}+\frac {x\,\left (16\,a^3\,c\,d\,e^3\,g^3-72\,a^2\,c^2\,d^2\,e^2\,f\,g^2+126\,a\,c^3\,d^3\,e\,f^2\,g+210\,c^4\,d^4\,f^3\right )}{315\,c^4\,d^4}+\frac {2\,g\,x^2\,\left (-2\,a^2\,e^2\,g^2+9\,a\,c\,d\,e\,f\,g+63\,c^2\,d^2\,f^2\right )}{105\,c^2\,d^2}+\frac {2\,g^2\,x^3\,\left (a\,e\,g+27\,c\,d\,f\right )}{63\,c\,d}\right )}{\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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