Optimal. Leaf size=281 \[ -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}-\frac {4 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^4 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.18, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {963, 79, 47, 37}
\begin {gather*} -\frac {4 g \sqrt {f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 \sqrt {d+e x} (e f-d g)^4}+\frac {2 \sqrt {f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 (d+e x)^{3/2} (e f-d g)^3}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{7 (d+e x)^{7/2} (e f-d g)}+\frac {2 \sqrt {f+g x} (2 c d (7 e f-4 d g)-e (-6 a e g-b d g+7 b e f))}{35 e^2 (d+e x)^{5/2} (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 963
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^{9/2} \sqrt {f+g x}} \, dx &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}-\frac {2 \int \frac {\frac {c d (7 e f-d g)-e (7 b e f-b d g-6 a e g)}{2 e^2}-\frac {7}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx}{7 (e f-d g)}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}-\frac {\left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx}{35 e^2 (e f-d g)^2}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}+\frac {\left (2 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right )\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{105 e^2 (e f-d g)^3}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac {2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt {f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac {2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}-\frac {4 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{105 e^2 (e f-d g)^4 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 301, normalized size = 1.07 \begin {gather*} -\frac {2 \sqrt {f+g x} \left (-105 c f^2 g (d+e x)^3+105 b f g^2 (d+e x)^3-105 a g^3 (d+e x)^3+35 c e f^2 (d+e x)^2 (f+g x)+70 c d f g (d+e x)^2 (f+g x)-70 b e f g (d+e x)^2 (f+g x)-35 b d g^2 (d+e x)^2 (f+g x)+105 a e g^2 (d+e x)^2 (f+g x)-42 c d e f (d+e x) (f+g x)^2+21 b e^2 f (d+e x) (f+g x)^2-21 c d^2 g (d+e x) (f+g x)^2+42 b d e g (d+e x) (f+g x)^2-63 a e^2 g (d+e x) (f+g x)^2+15 c d^2 e (f+g x)^3-15 b d e^2 (f+g x)^3+15 a e^3 (f+g x)^3\right )}{105 (e f-d g)^4 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 427, normalized size = 1.52
method | result | size |
default | \(\frac {2 \sqrt {g x +f}\, \left (48 a \,e^{3} g^{3} x^{3}+8 b d \,e^{2} g^{3} x^{3}-56 b \,e^{3} f \,g^{2} x^{3}+6 c \,d^{2} e \,g^{3} x^{3}-28 c d \,e^{2} f \,g^{2} x^{3}+70 c \,e^{3} f^{2} g \,x^{3}+168 a d \,e^{2} g^{3} x^{2}-24 a \,e^{3} f \,g^{2} x^{2}+28 b \,d^{2} e \,g^{3} x^{2}-200 b d \,e^{2} f \,g^{2} x^{2}+28 b \,e^{3} f^{2} g \,x^{2}+21 c \,d^{3} g^{3} x^{2}-101 c \,d^{2} e f \,g^{2} x^{2}+259 c d \,e^{2} f^{2} g \,x^{2}-35 c \,e^{3} f^{3} x^{2}+210 a \,d^{2} e \,g^{3} x -84 a d \,e^{2} f \,g^{2} x +18 a \,e^{3} f^{2} g x +35 b \,d^{3} g^{3} x -259 b \,d^{2} e f \,g^{2} x +101 b d \,e^{2} f^{2} g x -21 b \,e^{3} f^{3} x -28 c \,d^{3} f \,g^{2} x +200 c \,d^{2} e \,f^{2} g x -28 c d \,e^{2} f^{3} x +105 a \,d^{3} g^{3}-105 a \,d^{2} e f \,g^{2}+63 a d \,e^{2} f^{2} g -15 a \,e^{3} f^{3}-70 b \,d^{3} f \,g^{2}+28 b \,d^{2} e \,f^{2} g -6 b d \,e^{2} f^{3}+56 c \,d^{3} f^{2} g -8 c \,d^{2} e \,f^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (d g -e f \right )^{4}}\) | \(427\) |
gosper | \(\frac {2 \sqrt {g x +f}\, \left (48 a \,e^{3} g^{3} x^{3}+8 b d \,e^{2} g^{3} x^{3}-56 b \,e^{3} f \,g^{2} x^{3}+6 c \,d^{2} e \,g^{3} x^{3}-28 c d \,e^{2} f \,g^{2} x^{3}+70 c \,e^{3} f^{2} g \,x^{3}+168 a d \,e^{2} g^{3} x^{2}-24 a \,e^{3} f \,g^{2} x^{2}+28 b \,d^{2} e \,g^{3} x^{2}-200 b d \,e^{2} f \,g^{2} x^{2}+28 b \,e^{3} f^{2} g \,x^{2}+21 c \,d^{3} g^{3} x^{2}-101 c \,d^{2} e f \,g^{2} x^{2}+259 c d \,e^{2} f^{2} g \,x^{2}-35 c \,e^{3} f^{3} x^{2}+210 a \,d^{2} e \,g^{3} x -84 a d \,e^{2} f \,g^{2} x +18 a \,e^{3} f^{2} g x +35 b \,d^{3} g^{3} x -259 b \,d^{2} e f \,g^{2} x +101 b d \,e^{2} f^{2} g x -21 b \,e^{3} f^{3} x -28 c \,d^{3} f \,g^{2} x +200 c \,d^{2} e \,f^{2} g x -28 c d \,e^{2} f^{3} x +105 a \,d^{3} g^{3}-105 a \,d^{2} e f \,g^{2}+63 a d \,e^{2} f^{2} g -15 a \,e^{3} f^{3}-70 b \,d^{3} f \,g^{2}+28 b \,d^{2} e \,f^{2} g -6 b d \,e^{2} f^{3}+56 c \,d^{3} f^{2} g -8 c \,d^{2} e \,f^{3}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (g^{4} d^{4}-4 e \,g^{3} f \,d^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right )}\) | \(468\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 658 vs.
\(2 (267) = 534\).
time = 84.76, size = 658, normalized size = 2.34 \begin {gather*} \frac {2 \, {\left (21 \, c d^{3} g^{3} x^{2} + 56 \, c d^{3} f^{2} g - 70 \, b d^{3} f g^{2} + 105 \, a d^{3} g^{3} - 7 \, {\left (4 \, c d^{3} f g^{2} - 5 \, b d^{3} g^{3}\right )} x - {\left (15 \, a f^{3} - 2 \, {\left (35 \, c f^{2} g - 28 \, b f g^{2} + 24 \, a g^{3}\right )} x^{3} + {\left (35 \, c f^{3} - 28 \, b f^{2} g + 24 \, a f g^{2}\right )} x^{2} + 3 \, {\left (7 \, b f^{3} - 6 \, a f^{2} g\right )} x\right )} e^{3} - {\left (6 \, b d f^{3} - 63 \, a d f^{2} g + 4 \, {\left (7 \, c d f g^{2} - 2 \, b d g^{3}\right )} x^{3} - {\left (259 \, c d f^{2} g - 200 \, b d f g^{2} + 168 \, a d g^{3}\right )} x^{2} + {\left (28 \, c d f^{3} - 101 \, b d f^{2} g + 84 \, a d f g^{2}\right )} x\right )} e^{2} + {\left (6 \, c d^{2} g^{3} x^{3} - 8 \, c d^{2} f^{3} + 28 \, b d^{2} f^{2} g - 105 \, a d^{2} f g^{2} - {\left (101 \, c d^{2} f g^{2} - 28 \, b d^{2} g^{3}\right )} x^{2} + {\left (200 \, c d^{2} f^{2} g - 259 \, b d^{2} f g^{2} + 210 \, a d^{2} g^{3}\right )} x\right )} e\right )} \sqrt {g x + f} \sqrt {x e + d}}{105 \, {\left (d^{8} g^{4} + f^{4} x^{4} e^{8} - 4 \, {\left (d f^{3} g x^{4} - d f^{4} x^{3}\right )} e^{7} + 2 \, {\left (3 \, d^{2} f^{2} g^{2} x^{4} - 8 \, d^{2} f^{3} g x^{3} + 3 \, d^{2} f^{4} x^{2}\right )} e^{6} - 4 \, {\left (d^{3} f g^{3} x^{4} - 6 \, d^{3} f^{2} g^{2} x^{3} + 6 \, d^{3} f^{3} g x^{2} - d^{3} f^{4} x\right )} e^{5} + {\left (d^{4} g^{4} x^{4} - 16 \, d^{4} f g^{3} x^{3} + 36 \, d^{4} f^{2} g^{2} x^{2} - 16 \, d^{4} f^{3} g x + d^{4} f^{4}\right )} e^{4} + 4 \, {\left (d^{5} g^{4} x^{3} - 6 \, d^{5} f g^{3} x^{2} + 6 \, d^{5} f^{2} g^{2} x - d^{5} f^{3} g\right )} e^{3} + 2 \, {\left (3 \, d^{6} g^{4} x^{2} - 8 \, d^{6} f g^{3} x + 3 \, d^{6} f^{2} g^{2}\right )} e^{2} + 4 \, {\left (d^{7} g^{4} x - d^{7} f g^{3}\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. 752 vs.
\(2 (267) = 534\).
time = 2.51, size = 752, normalized size = 2.68 \begin {gather*} \frac {2 \, {\left ({\left ({\left (g x + f\right )} {\left (\frac {2 \, {\left (3 \, c d^{2} g^{10} e^{4} - 14 \, c d f g^{9} e^{5} + 4 \, b d g^{10} e^{5} + 35 \, c f^{2} g^{8} e^{6} - 28 \, b f g^{9} e^{6} + 24 \, a g^{10} e^{6}\right )} {\left (g x + f\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}} + \frac {7 \, {\left (3 \, c d^{3} g^{11} e^{3} - 17 \, c d^{2} f g^{10} e^{4} + 4 \, b d^{2} g^{11} e^{4} + 49 \, c d f^{2} g^{9} e^{5} - 32 \, b d f g^{10} e^{5} + 24 \, a d g^{11} e^{5} - 35 \, c f^{3} g^{8} e^{6} + 28 \, b f^{2} g^{9} e^{6} - 24 \, a f g^{10} e^{6}\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}}\right )} - \frac {35 \, {\left (2 \, c d^{3} f g^{11} e^{3} - b d^{3} g^{12} e^{3} - 12 \, c d^{2} f^{2} g^{10} e^{4} + 9 \, b d^{2} f g^{11} e^{4} - 6 \, a d^{2} g^{12} e^{4} + 18 \, c d f^{3} g^{9} e^{5} - 15 \, b d f^{2} g^{10} e^{5} + 12 \, a d f g^{11} e^{5} - 8 \, c f^{4} g^{8} e^{6} + 7 \, b f^{3} g^{9} e^{6} - 6 \, a f^{2} g^{10} e^{6}\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}}\right )} {\left (g x + f\right )} + \frac {105 \, {\left (c d^{3} f^{2} g^{11} e^{3} - b d^{3} f g^{12} e^{3} + a d^{3} g^{13} e^{3} - 3 \, c d^{2} f^{3} g^{10} e^{4} + 3 \, b d^{2} f^{2} g^{11} e^{4} - 3 \, a d^{2} f g^{12} e^{4} + 3 \, c d f^{4} g^{9} e^{5} - 3 \, b d f^{3} g^{10} e^{5} + 3 \, a d f^{2} g^{11} e^{5} - c f^{5} g^{8} e^{6} + b f^{4} g^{9} e^{6} - a f^{3} g^{10} e^{6}\right )}}{d^{4} g^{6} {\left | g \right |} e^{3} - 4 \, d^{3} f g^{5} {\left | g \right |} e^{4} + 6 \, d^{2} f^{2} g^{4} {\left | g \right |} e^{5} - 4 \, d f^{3} g^{3} {\left | g \right |} e^{6} + f^{4} g^{2} {\left | g \right |} e^{7}}\right )} \sqrt {g x + f}}{105 \, {\left (d g^{2} + {\left (g x + f\right )} g e - f g e\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.65, size = 452, normalized size = 1.61 \begin {gather*} \frac {\sqrt {f+g\,x}\,\left (\frac {x^3\,\left (12\,c\,d^2\,e\,g^3-56\,c\,d\,e^2\,f\,g^2+16\,b\,d\,e^2\,g^3+140\,c\,e^3\,f^2\,g-112\,b\,e^3\,f\,g^2+96\,a\,e^3\,g^3\right )}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}-\frac {-112\,c\,d^3\,f^2\,g+140\,b\,d^3\,f\,g^2-210\,a\,d^3\,g^3+16\,c\,d^2\,e\,f^3-56\,b\,d^2\,e\,f^2\,g+210\,a\,d^2\,e\,f\,g^2+12\,b\,d\,e^2\,f^3-126\,a\,d\,e^2\,f^2\,g+30\,a\,e^3\,f^3}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}+\frac {x\,\left (-56\,c\,d^3\,f\,g^2+70\,b\,d^3\,g^3+400\,c\,d^2\,e\,f^2\,g-518\,b\,d^2\,e\,f\,g^2+420\,a\,d^2\,e\,g^3-56\,c\,d\,e^2\,f^3+202\,b\,d\,e^2\,f^2\,g-168\,a\,d\,e^2\,f\,g^2-42\,b\,e^3\,f^3+36\,a\,e^3\,f^2\,g\right )}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}+\frac {2\,x^2\,\left (7\,d\,g-e\,f\right )\,\left (3\,c\,d^2\,g^2-14\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+35\,c\,e^2\,f^2-28\,b\,e^2\,f\,g+24\,a\,e^2\,g^2\right )}{105\,e^3\,{\left (d\,g-e\,f\right )}^4}\right )}{x^3\,\sqrt {d+e\,x}+\frac {d^3\,\sqrt {d+e\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {d+e\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {d+e\,x}}{e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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