Optimal. Leaf size=193 \[ -\frac {4 (2 c d-b e) (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{315 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}} \]
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Rubi [A]
time = 0.19, antiderivative size = 193, 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 = {808, 670, 662}
\begin {gather*} -\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rule 808
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {1}{2} \left (c e^3 f-\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{9 c e^3}\\ &=-\frac {2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}}+\frac {(2 (2 c d-b e) (9 c e f-c d g-4 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{63 c^2 e}\\ &=-\frac {4 (2 c d-b e) (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{315 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 121, normalized size = 0.63 \begin {gather*} -\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (8 b^2 e^2 g-2 b c e (9 e f+17 d g+10 e g x)+c^2 \left (26 d^2 g+5 e^2 x (9 f+7 g x)+d e (81 f+65 g x)\right )\right )}{315 c^3 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 133, normalized size = 0.69
method | result | size |
default | \(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{2} \left (35 g \,x^{2} c^{2} e^{2}-20 b c \,e^{2} g x +65 c^{2} d e g x +45 c^{2} e^{2} f x +8 b^{2} e^{2} g -34 b c d e g -18 b c \,e^{2} f +26 c^{2} d^{2} g +81 c^{2} d e f \right )}{315 \sqrt {e x +d}\, c^{3} e^{2}}\) | \(133\) |
gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (35 g \,x^{2} c^{2} e^{2}-20 b c \,e^{2} g x +65 c^{2} d e g x +45 c^{2} e^{2} f x +8 b^{2} e^{2} g -34 b c d e g -18 b c \,e^{2} f +26 c^{2} d^{2} g +81 c^{2} d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{315 c^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 307, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (5 \, c^{3} x^{3} e^{3} + 9 \, c^{3} d^{3} - 20 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - {\left (c^{3} d e^{2} - 8 \, b c^{2} e^{3}\right )} x^{2} - {\left (13 \, c^{3} d^{2} e - 12 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x\right )} \sqrt {-c x e + c d - b e} f e^{\left (-1\right )}}{35 \, c^{2}} - \frac {2 \, {\left (35 \, c^{4} x^{4} e^{4} + 26 \, c^{4} d^{4} - 86 \, b c^{3} d^{3} e + 102 \, b^{2} c^{2} d^{2} e^{2} - 50 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} - 5 \, {\left (c^{4} d e^{3} - 10 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (23 \, c^{4} d^{2} e^{2} - 22 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} + {\left (13 \, c^{4} d^{3} e - 30 \, b c^{3} d^{2} e^{2} + 21 \, b^{2} c^{2} d e^{3} - 4 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c x e + c d - b e} g e^{\left (-2\right )}}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.96, size = 319, normalized size = 1.65 \begin {gather*} -\frac {2 \, {\left (26 \, c^{4} d^{4} g + {\left (35 \, c^{4} g x^{4} - 18 \, b^{3} c f + 8 \, b^{4} g + 5 \, {\left (9 \, c^{4} f + 10 \, b c^{3} g\right )} x^{3} + 3 \, {\left (24 \, b c^{3} f + b^{2} c^{2} g\right )} x^{2} + {\left (9 \, b^{2} c^{2} f - 4 \, b^{3} c g\right )} x\right )} e^{4} - {\left (5 \, c^{4} d g x^{3} - 117 \, b^{2} c^{2} d f + 50 \, b^{3} c d g + 3 \, {\left (3 \, c^{4} d f - 22 \, b c^{3} d g\right )} x^{2} - 3 \, {\left (36 \, b c^{3} d f + 7 \, b^{2} c^{2} d g\right )} x\right )} e^{3} - 3 \, {\left (23 \, c^{4} d^{2} g x^{2} + 60 \, b c^{3} d^{2} f - 34 \, b^{2} c^{2} d^{2} g + {\left (39 \, c^{4} d^{2} f + 10 \, b c^{3} d^{2} g\right )} x\right )} e^{2} + {\left (13 \, c^{4} d^{3} g x + 81 \, c^{4} d^{3} f - 86 \, b c^{3} d^{3} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{315 \, {\left (c^{3} x e^{3} + c^{3} d e^{2}\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 (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\sqrt {d + e 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. 1818 vs.
\(2 (175) = 350\).
time = 1.19, size = 1818, normalized size = 9.42 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.05, size = 239, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e\,x^3\,\left (10\,b\,e\,g-c\,d\,g+9\,c\,e\,f\right )}{63}+\frac {2\,x^2\,\left (g\,b^2\,e^2+22\,g\,b\,c\,d\,e+24\,f\,b\,c\,e^2-23\,g\,c^2\,d^2-3\,f\,c^2\,d\,e\right )}{105\,c}+\frac {2\,c\,e^2\,g\,x^4}{9}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\left (8\,g\,b^2\,e^2-34\,g\,b\,c\,d\,e-18\,f\,b\,c\,e^2+26\,g\,c^2\,d^2+81\,f\,c^2\,d\,e\right )}{315\,c^3\,e^2}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\left (-4\,g\,b^2\,e^2+17\,g\,b\,c\,d\,e+9\,f\,b\,c\,e^2-13\,g\,c^2\,d^2+117\,f\,c^2\,d\,e\right )}{315\,c^2\,e}\right )}{\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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