Optimal. Leaf size=198 \[ -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}+\frac {2 (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)) \sqrt {f+g x}}{15 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 \left (2 e g (5 b e f-b d g-4 a e g)-c \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{15 e^2 (e f-d g)^3 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.13, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {963, 79, 37}
\begin {gather*} \frac {2 \sqrt {f+g x} \left (2 e g (-4 a e g-b d g+5 b e f)-c \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{15 e^2 \sqrt {d+e x} (e f-d g)^3}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{5 (d+e x)^{5/2} (e f-d g)}+\frac {2 \sqrt {f+g x} (2 c d (5 e f-3 d g)-e (-4 a e g-b d g+5 b e f))}{15 e^2 (d+e x)^{3/2} (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rule 963
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^{7/2} \sqrt {f+g x}} \, dx &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {c d (5 e f-d g)-e (5 b e f-b d g-4 a e g)}{2 e^2}-\frac {5}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx}{5 (e f-d g)}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}+\frac {2 (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)) \sqrt {f+g x}}{15 e^2 (e f-d g)^2 (d+e x)^{3/2}}-\frac {\left (2 e g (5 b e f-b d g-4 a e g)-c \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right )\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{15 e^2 (e f-d g)^2}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{5 (e f-d g) (d+e x)^{5/2}}+\frac {2 (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)) \sqrt {f+g x}}{15 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 \left (2 e g (5 b e f-b d g-4 a e g)-c \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right )\right ) \sqrt {f+g x}}{15 e^2 (e f-d g)^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 177, normalized size = 0.89 \begin {gather*} -\frac {2 \sqrt {f+g x} \left (15 c f^2-15 b f g+15 a g^2-\frac {10 c d f (f+g x)}{d+e x}+\frac {5 b e f (f+g x)}{d+e x}+\frac {5 b d g (f+g x)}{d+e x}-\frac {10 a e g (f+g x)}{d+e x}+\frac {3 c d^2 (f+g x)^2}{(d+e x)^2}-\frac {3 b d e (f+g x)^2}{(d+e x)^2}+\frac {3 a e^2 (f+g x)^2}{(d+e x)^2}\right )}{15 (e f-d g)^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 210, normalized size = 1.06
method | result | size |
default | \(\frac {2 \sqrt {g x +f}\, \left (8 a \,e^{2} g^{2} x^{2}+2 b d e \,g^{2} x^{2}-10 b \,e^{2} f g \,x^{2}+3 c \,d^{2} g^{2} x^{2}-10 c d e f g \,x^{2}+15 c \,e^{2} f^{2} x^{2}+20 a d e \,g^{2} x -4 a \,e^{2} f g x +5 b \,d^{2} g^{2} x -26 b d e f g x +5 b \,e^{2} f^{2} x -4 c \,d^{2} f g x +20 c d e \,f^{2} x +15 a \,d^{2} g^{2}-10 a d e f g +3 a \,e^{2} f^{2}-10 b \,d^{2} f g +2 b d e \,f^{2}+8 c \,d^{2} f^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (d g -e f \right )^{3}}\) | \(210\) |
gosper | \(\frac {2 \sqrt {g x +f}\, \left (8 a \,e^{2} g^{2} x^{2}+2 b d e \,g^{2} x^{2}-10 b \,e^{2} f g \,x^{2}+3 c \,d^{2} g^{2} x^{2}-10 c d e f g \,x^{2}+15 c \,e^{2} f^{2} x^{2}+20 a d e \,g^{2} x -4 a \,e^{2} f g x +5 b \,d^{2} g^{2} x -26 b d e f g x +5 b \,e^{2} f^{2} x -4 c \,d^{2} f g x +20 c d e \,f^{2} x +15 a \,d^{2} g^{2}-10 a d e f g +3 a \,e^{2} f^{2}-10 b \,d^{2} f g +2 b d e \,f^{2}+8 c \,d^{2} f^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (g^{3} d^{3}-3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g -e^{3} f^{3}\right )}\) | \(238\) |
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 [A]
time = 19.89, size = 360, normalized size = 1.82 \begin {gather*} \frac {2 \, {\left (3 \, c d^{2} g^{2} x^{2} + 8 \, c d^{2} f^{2} - 10 \, b d^{2} f g + 15 \, a d^{2} g^{2} - {\left (4 \, c d^{2} f g - 5 \, b d^{2} g^{2}\right )} x + {\left (3 \, a f^{2} + {\left (15 \, c f^{2} - 10 \, b f g + 8 \, a g^{2}\right )} x^{2} + {\left (5 \, b f^{2} - 4 \, a f g\right )} x\right )} e^{2} + 2 \, {\left (b d f^{2} - 5 \, a d f g - {\left (5 \, c d f g - b d g^{2}\right )} x^{2} + {\left (10 \, c d f^{2} - 13 \, b d f g + 10 \, a d g^{2}\right )} x\right )} e\right )} \sqrt {g x + f} \sqrt {x e + d}}{15 \, {\left (d^{6} g^{3} - f^{3} x^{3} e^{6} + 3 \, {\left (d f^{2} g x^{3} - d f^{3} x^{2}\right )} e^{5} - 3 \, {\left (d^{2} f g^{2} x^{3} - 3 \, d^{2} f^{2} g x^{2} + d^{2} f^{3} x\right )} e^{4} + {\left (d^{3} g^{3} x^{3} - 9 \, d^{3} f g^{2} x^{2} + 9 \, d^{3} f^{2} g x - d^{3} f^{3}\right )} e^{3} + 3 \, {\left (d^{4} g^{3} x^{2} - 3 \, d^{4} f g^{2} x + d^{4} f^{2} g\right )} e^{2} + 3 \, {\left (d^{5} g^{3} x - d^{5} f g^{2}\right )} e\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 {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {7}{2}} \sqrt {f + g 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. 452 vs.
\(2 (188) = 376\).
time = 6.15, size = 452, normalized size = 2.28 \begin {gather*} \frac {2 \, {\left ({\left (g x + f\right )} {\left (\frac {{\left (3 \, c d^{2} g^{8} e^{2} - 10 \, c d f g^{7} e^{3} + 2 \, b d g^{8} e^{3} + 15 \, c f^{2} g^{6} e^{4} - 10 \, b f g^{7} e^{4} + 8 \, a g^{8} e^{4}\right )} {\left (g x + f\right )}}{d^{3} g^{5} {\left | g \right |} e^{2} - 3 \, d^{2} f g^{4} {\left | g \right |} e^{3} + 3 \, d f^{2} g^{3} {\left | g \right |} e^{4} - f^{3} g^{2} {\left | g \right |} e^{5}} - \frac {5 \, {\left (2 \, c d^{2} f g^{8} e^{2} - b d^{2} g^{9} e^{2} - 8 \, c d f^{2} g^{7} e^{3} + 6 \, b d f g^{8} e^{3} - 4 \, a d g^{9} e^{3} + 6 \, c f^{3} g^{6} e^{4} - 5 \, b f^{2} g^{7} e^{4} + 4 \, a f g^{8} e^{4}\right )}}{d^{3} g^{5} {\left | g \right |} e^{2} - 3 \, d^{2} f g^{4} {\left | g \right |} e^{3} + 3 \, d f^{2} g^{3} {\left | g \right |} e^{4} - f^{3} g^{2} {\left | g \right |} e^{5}}\right )} + \frac {15 \, {\left (c d^{2} f^{2} g^{8} e^{2} - b d^{2} f g^{9} e^{2} + a d^{2} g^{10} e^{2} - 2 \, c d f^{3} g^{7} e^{3} + 2 \, b d f^{2} g^{8} e^{3} - 2 \, a d f g^{9} e^{3} + c f^{4} g^{6} e^{4} - b f^{3} g^{7} e^{4} + a f^{2} g^{8} e^{4}\right )}}{d^{3} g^{5} {\left | g \right |} e^{2} - 3 \, d^{2} f g^{4} {\left | g \right |} e^{3} + 3 \, d f^{2} g^{3} {\left | g \right |} e^{4} - f^{3} g^{2} {\left | g \right |} e^{5}}\right )} \sqrt {g x + f}}{15 \, {\left (d g^{2} + {\left (g x + f\right )} g e - f g e\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.30, size = 260, normalized size = 1.31 \begin {gather*} \frac {\sqrt {f+g\,x}\,\left (\frac {16\,c\,d^2\,f^2-20\,b\,d^2\,f\,g+30\,a\,d^2\,g^2+4\,b\,d\,e\,f^2-20\,a\,d\,e\,f\,g+6\,a\,e^2\,f^2}{15\,e^2\,{\left (d\,g-e\,f\right )}^3}+\frac {x\,\left (-8\,c\,d^2\,f\,g+10\,b\,d^2\,g^2+40\,c\,d\,e\,f^2-52\,b\,d\,e\,f\,g+40\,a\,d\,e\,g^2+10\,b\,e^2\,f^2-8\,a\,e^2\,f\,g\right )}{15\,e^2\,{\left (d\,g-e\,f\right )}^3}+\frac {x^2\,\left (6\,c\,d^2\,g^2-20\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+30\,c\,e^2\,f^2-20\,b\,e^2\,f\,g+16\,a\,e^2\,g^2\right )}{15\,e^2\,{\left (d\,g-e\,f\right )}^3}\right )}{x^2\,\sqrt {d+e\,x}+\frac {d^2\,\sqrt {d+e\,x}}{e^2}+\frac {2\,d\,x\,\sqrt {d+e\,x}}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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