Optimal. Leaf size=200 \[ -\frac {8 (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac {2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac {8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}} \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)^2 \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}} \, dx &=\frac {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}}{9 c d (d+e x)^{5/2}}+\frac {\left (4 \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) \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}} \, dx}{9 c d e^2}\\ &=\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{5/2}}-\frac {\left (4 (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right )\right ) \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}} \, dx}{63 c^2 d^2 e}\\ &=-\frac {8 (c d f-a e g) \left (2 a e^2 g-c d (7 e f-5 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac {8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac {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}}{9 c d (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 90, normalized size = 0.45 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (8 a^2 e^2 g^2-4 a c d e g (9 f+5 g x)+c^2 d^2 \left (63 f^2+90 f g x+35 g^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 108, normalized size = 0.54
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2} \left (35 g^{2} x^{2} c^{2} d^{2}-20 a c d e \,g^{2} x +90 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-36 a c d e f g +63 f^{2} c^{2} d^{2}\right )}{315 \sqrt {e x +d}\, c^{3} d^{3}}\) | \(108\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (35 g^{2} x^{2} c^{2} d^{2}-20 a c d e \,g^{2} x +90 c^{2} d^{2} f g x +8 a^{2} e^{2} g^{2}-36 a c d e f g +63 f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 192, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d x e + a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{2}}{5 \, c d} + \frac {4 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} x^{2} e + a^{2} c d x e^{2} - 2 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f g}{35 \, c^{2} d^{2}} + \frac {2 \, {\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} x^{3} e + 3 \, a^{2} c^{2} d^{2} x^{2} e^{2} - 4 \, a^{3} c d x e^{3} + 8 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} g^{2}}{315 \, c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.05, size = 229, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} d^{4} g^{2} x^{4} + 90 \, c^{4} d^{4} f g x^{3} + 63 \, c^{4} d^{4} f^{2} x^{2} + 8 \, a^{4} g^{2} e^{4} - 4 \, {\left (a^{3} c d g^{2} x + 9 \, a^{3} c d f g\right )} e^{3} + 3 \, {\left (a^{2} c^{2} d^{2} g^{2} x^{2} + 6 \, a^{2} c^{2} d^{2} f g x + 21 \, a^{2} c^{2} d^{2} f^{2}\right )} e^{2} + 2 \, {\left (25 \, a c^{3} d^{3} g^{2} x^{3} + 72 \, a c^{3} d^{3} f g x^{2} + 63 \, a c^{3} d^{3} f^{2} x\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^{3} d^{3} x e + c^{3} d^{4}\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 {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )^{2}}{\left (d + e x\right )^{\frac {3}{2}}}\, 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. 1119 vs.
\(2 (187) = 374\).
time = 3.11, size = 1119, normalized size = 5.60 \begin {gather*} \frac {2}{315} \, {\left (6 \, c d f g {\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 )} - c d g^{2} {\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 )} - 21 \, c d f^{2} {\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 )} - 42 \, a f 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 (-1\right )} + 105 \, a f^{2} {\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 )} + 3 \, a 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 )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.43, size = 206, normalized size = 1.03 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g\,x^3\,\left (5\,a\,e\,g+9\,c\,d\,f\right )}{63}+\frac {16\,a^4\,e^4\,g^2-72\,a^3\,c\,d\,e^3\,f\,g+126\,a^2\,c^2\,d^2\,e^2\,f^2}{315\,c^3\,d^3}+\frac {x^2\,\left (6\,a^2\,c^2\,d^2\,e^2\,g^2+288\,a\,c^3\,d^3\,e\,f\,g+126\,c^4\,d^4\,f^2\right )}{315\,c^3\,d^3}+\frac {2\,c\,d\,g^2\,x^4}{9}+\frac {4\,a\,e\,x\,\left (-2\,a^2\,e^2\,g^2+9\,a\,c\,d\,e\,f\,g+63\,c^2\,d^2\,f^2\right )}{315\,c^2\,d^2}\right )}{\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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