Optimal. Leaf size=321 \[ \frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^4 d^4 e g \sqrt {d+e x}}-\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 e}-\frac {2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}} \]
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Rubi [A]
time = 0.27, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {894, 884, 808,
662} \begin {gather*} \frac {8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt {d+e x}}-\frac {8 \sqrt {d+e 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) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}-\frac {2 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 808
Rule 884
Rule 894
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}-\frac {1}{7} \left (-7 d+\frac {6 a e^2}{c d}+\frac {e f}{g}\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\\ &=-\frac {2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}-\frac {\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )\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}{35 c^2 d^2 g}\\ &=-\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 e}-\frac {2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}+\frac {\left (4 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \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}{105 c^3 d^3 e g}\\ &=\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \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}}{105 c^4 d^4 e g \sqrt {d+e x}}-\frac {8 (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 e}-\frac {2 \left (6 a e^2 g+c d (e f-7 d g)\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 g \sqrt {d+e x}}+\frac {2 e (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d g \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 169, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-48 a^3 e^4 g^2+8 a^2 c d e^2 g (14 e f+7 d g+3 e g x)-2 a c^2 d^2 e \left (14 d g (5 f+g x)+e \left (35 f^2+28 f g x+9 g^2 x^2\right )\right )+c^3 d^3 \left (7 d \left (15 f^2+10 f g x+3 g^2 x^2\right )+e x \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )\right )}{105 c^4 d^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 237, normalized size = 0.74
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-15 g^{2} e \,x^{3} c^{3} d^{3}+18 a \,c^{2} d^{2} e^{2} g^{2} x^{2}-21 c^{3} d^{4} g^{2} x^{2}-42 c^{3} d^{3} e f g \,x^{2}-24 a^{2} c d \,e^{3} g^{2} x +28 a \,c^{2} d^{3} e \,g^{2} x +56 a \,c^{2} d^{2} e^{2} f g x -70 c^{3} d^{4} f g x -35 c^{3} d^{3} e \,f^{2} x +48 a^{3} e^{4} g^{2}-56 a^{2} c \,d^{2} e^{2} g^{2}-112 a^{2} c d \,e^{3} f g +140 a \,c^{2} d^{3} e f g +70 a \,c^{2} d^{2} e^{2} f^{2}-105 d^{4} f^{2} c^{3}\right )}{105 \sqrt {e x +d}\, c^{4} d^{4}}\) | \(237\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-15 g^{2} e \,x^{3} c^{3} d^{3}+18 a \,c^{2} d^{2} e^{2} g^{2} x^{2}-21 c^{3} d^{4} g^{2} x^{2}-42 c^{3} d^{3} e f g \,x^{2}-24 a^{2} c d \,e^{3} g^{2} x +28 a \,c^{2} d^{3} e \,g^{2} x +56 a \,c^{2} d^{2} e^{2} f g x -70 c^{3} d^{4} f g x -35 c^{3} d^{3} e \,f^{2} x +48 a^{3} e^{4} g^{2}-56 a^{2} c \,d^{2} e^{2} g^{2}-112 a^{2} c d \,e^{3} f g +140 a \,c^{2} d^{3} e f g +70 a \,c^{2} d^{2} e^{2} f^{2}-105 d^{4} f^{2} c^{3}\right ) \sqrt {e x +d}}{105 c^{4} d^{4} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(255\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 306, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} e + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{2}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {4 \, {\left (3 \, c^{3} d^{3} x^{3} e - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (15 \, c^{4} d^{4} x^{4} e + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \, {\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} - {\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} g^{2}}{105 \, \sqrt {c d x + a e} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.53, size = 254, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (21 \, c^{3} d^{4} g^{2} x^{2} + 70 \, c^{3} d^{4} f g x + 105 \, c^{3} d^{4} f^{2} - 48 \, a^{3} g^{2} e^{4} + 8 \, {\left (3 \, a^{2} c d g^{2} x + 14 \, a^{2} c d f g\right )} e^{3} - 2 \, {\left (9 \, a c^{2} d^{2} g^{2} x^{2} + 28 \, a c^{2} d^{2} f g x + 35 \, a c^{2} d^{2} f^{2} - 28 \, a^{2} c d^{2} g^{2}\right )} e^{2} + {\left (15 \, c^{3} d^{3} g^{2} x^{3} + 42 \, c^{3} d^{3} f g x^{2} - 140 \, a c^{2} d^{3} f g + 7 \, {\left (5 \, c^{3} d^{3} f^{2} - 4 \, a c^{2} d^{3} g^{2}\right )} 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}}{105 \, {\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 {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )^{2}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, 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. 700 vs.
\(2 (307) = 614\).
time = 3.54, size = 700, normalized size = 2.18 \begin {gather*} \frac {2 \, {\left (c^{3} d^{4} f^{2} - 2 \, a c^{2} d^{3} f g e - a c^{2} d^{2} f^{2} e^{2} + a^{2} c d^{2} g^{2} e^{2} + 2 \, a^{2} c d f g e^{3} - a^{3} g^{2} e^{4}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{4} d^{4}} - \frac {4 \, {\left (3 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} g^{2} - 14 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{5} f g e + 35 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{4} f^{2} e^{2} + 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} g^{2} e^{2} - 42 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} f g e^{3} - 35 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{2} f^{2} e^{4} + 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} g^{2} e^{4} + 56 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d f g e^{5} - 24 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} g^{2} e^{6}\right )} e^{\left (-3\right )}}{105 \, c^{4} d^{4}} + \frac {2 \, {\left (70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{3} f g e^{3} + 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} f^{2} e^{4} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d^{2} g^{2} e^{4} + 21 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d^{2} g^{2} e - 140 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d f g e^{5} + 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d f g e^{2} + 105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} g^{2} e^{6} - 63 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a g^{2} e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} g^{2}\right )} e^{\left (-6\right )}}{105 \, c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.71, size = 279, normalized size = 0.87 \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^2\,x^3\,\sqrt {d+e\,x}}{7\,c\,d}-\frac {\sqrt {d+e\,x}\,\left (96\,a^3\,e^4\,g^2-112\,a^2\,c\,d^2\,e^2\,g^2-224\,a^2\,c\,d\,e^3\,f\,g+280\,a\,c^2\,d^3\,e\,f\,g+140\,a\,c^2\,d^2\,e^2\,f^2-210\,c^3\,d^4\,f^2\right )}{105\,c^4\,d^4\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,a^2\,c\,d\,e^3\,g^2-56\,a\,c^2\,d^3\,e\,g^2-112\,a\,c^2\,d^2\,e^2\,f\,g+140\,c^3\,d^4\,f\,g+70\,c^3\,d^3\,e\,f^2\right )}{105\,c^4\,d^4\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}\,\left (7\,c\,g\,d^2+14\,c\,f\,d\,e-6\,a\,g\,e^2\right )}{35\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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