Optimal. Leaf size=281 \[ -\frac {4 g \sqrt {f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 \sqrt {d+e x} (e f-d g)^4}+\frac {2 \sqrt {f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 (d+e x)^{3/2} (e f-d g)^3}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{7 (d+e x)^{7/2} (e f-d g)}+\frac {2 \sqrt {f+g x} (2 c d (7 e f-4 d g)-e (-6 a e g-b d g+7 b e f))}{35 e^2 (d+e x)^{5/2} (e f-d g)^2} \]
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Rubi [A] time = 0.29, 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 = {949, 78, 45, 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 45
Rule 78
Rule 949
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 = 332, normalized size = 1.18 \begin {gather*} \frac {2 \sqrt {f+g x} \left (3 a \left (35 d^3 g^3-35 d^2 e g^2 (f-2 g x)+7 d e^2 g \left (3 f^2-4 f g x+8 g^2 x^2\right )+e^3 \left (-5 f^3+6 f^2 g x-8 f g^2 x^2+16 g^3 x^3\right )\right )+b \left (35 d^3 g^2 (g x-2 f)+7 d^2 e g \left (4 f^2-37 f g x+4 g^2 x^2\right )+d e^2 \left (-6 f^3+101 f^2 g x-200 f g^2 x^2+8 g^3 x^3\right )-7 e^3 f x \left (3 f^2-4 f g x+8 g^2 x^2\right )\right )+c \left (7 d^3 g \left (8 f^2-4 f g x+3 g^2 x^2\right )+d^2 e \left (-8 f^3+200 f^2 g x-101 f g^2 x^2+6 g^3 x^3\right )-7 d e^2 f x \left (4 f^2-37 f g x+4 g^2 x^2\right )-35 e^3 f^2 x^2 (f-2 g x)\right )\right )}{105 (d+e x)^{7/2} (e f-d g)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 311, normalized size = 1.11 \begin {gather*} -\frac {2 \sqrt {f+g x} \left (\frac {15 a e^3 (f+g x)^3}{(d+e x)^3}-\frac {63 a e^2 g (f+g x)^2}{(d+e x)^2}+\frac {105 a e g^2 (f+g x)}{d+e x}-105 a g^3-\frac {15 b d e^2 (f+g x)^3}{(d+e x)^3}+\frac {21 b e^2 f (f+g x)^2}{(d+e x)^2}-\frac {35 b d g^2 (f+g x)}{d+e x}+\frac {42 b d e g (f+g x)^2}{(d+e x)^2}-\frac {70 b e f g (f+g x)}{d+e x}+105 b f g^2-\frac {21 c d^2 g (f+g x)^2}{(d+e x)^2}+\frac {15 c d^2 e (f+g x)^3}{(d+e x)^3}+\frac {35 c e f^2 (f+g x)}{d+e x}+\frac {70 c d f g (f+g x)}{d+e x}-\frac {42 c d e f (f+g x)^2}{(d+e x)^2}-105 c f^2 g\right )}{105 \sqrt {d+e x} (e f-d g)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 36.00, size = 641, normalized size = 2.28 \begin {gather*} \frac {2 \, {\left (105 \, a d^{3} g^{3} - {\left (8 \, c d^{2} e + 6 \, b d e^{2} + 15 \, a e^{3}\right )} f^{3} + 7 \, {\left (8 \, c d^{3} + 4 \, b d^{2} e + 9 \, a d e^{2}\right )} f^{2} g - 35 \, {\left (2 \, b d^{3} + 3 \, a d^{2} e\right )} f g^{2} + 2 \, {\left (35 \, c e^{3} f^{2} g - 14 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f g^{2} + {\left (3 \, c d^{2} e + 4 \, b d e^{2} + 24 \, a e^{3}\right )} g^{3}\right )} x^{3} - {\left (35 \, c e^{3} f^{3} - 7 \, {\left (37 \, c d e^{2} + 4 \, b e^{3}\right )} f^{2} g + {\left (101 \, c d^{2} e + 200 \, b d e^{2} + 24 \, a e^{3}\right )} f g^{2} - 7 \, {\left (3 \, c d^{3} + 4 \, b d^{2} e + 24 \, a d e^{2}\right )} g^{3}\right )} x^{2} - {\left (7 \, {\left (4 \, c d e^{2} + 3 \, b e^{3}\right )} f^{3} - {\left (200 \, c d^{2} e + 101 \, b d e^{2} + 18 \, a e^{3}\right )} f^{2} g + 7 \, {\left (4 \, c d^{3} + 37 \, b d^{2} e + 12 \, a d e^{2}\right )} f g^{2} - 35 \, {\left (b d^{3} + 6 \, a d^{2} e\right )} g^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{105 \, {\left (d^{4} e^{4} f^{4} - 4 \, d^{5} e^{3} f^{3} g + 6 \, d^{6} e^{2} f^{2} g^{2} - 4 \, d^{7} e f g^{3} + d^{8} g^{4} + {\left (e^{8} f^{4} - 4 \, d e^{7} f^{3} g + 6 \, d^{2} e^{6} f^{2} g^{2} - 4 \, d^{3} e^{5} f g^{3} + d^{4} e^{4} g^{4}\right )} x^{4} + 4 \, {\left (d e^{7} f^{4} - 4 \, d^{2} e^{6} f^{3} g + 6 \, d^{3} e^{5} f^{2} g^{2} - 4 \, d^{4} e^{4} f g^{3} + d^{5} e^{3} g^{4}\right )} x^{3} + 6 \, {\left (d^{2} e^{6} f^{4} - 4 \, d^{3} e^{5} f^{3} g + 6 \, d^{4} e^{4} f^{2} g^{2} - 4 \, d^{5} e^{3} f g^{3} + d^{6} e^{2} g^{4}\right )} x^{2} + 4 \, {\left (d^{3} e^{5} f^{4} - 4 \, d^{4} e^{4} f^{3} g + 6 \, d^{5} e^{3} f^{2} g^{2} - 4 \, d^{6} e^{2} f g^{3} + d^{7} e g^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 1868, normalized size = 6.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 468, normalized size = 1.67 \begin {gather*} \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 )} \end {gather*}
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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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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sympy [F(-1)] 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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