Optimal. Leaf size=231 \[ -\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (c e f+7 c d g-4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 231, 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 = {806, 676, 674,
214} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 674
Rule 676
Rule 806
Rubi steps
\begin {align*} \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {(c e f+7 c d g-4 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(c (c e f+7 c d g-4 b e g)) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e (2 c d-b e)}\\ &=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(c (c e f+7 c d g-4 b e g)) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )}{4 (2 c d-b e)}\\ &=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (c e f+7 c d g-4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 181, normalized size = 0.78 \begin {gather*} \frac {c \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {2 b e (d g+e (f+2 g x))+c \left (-5 d^2 g+e^2 f x-3 d e (f+3 g x)\right )}{c (2 c d-b e) (d+e x)^2}+\frac {(c e f+7 c d g-4 b e g) \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{3/2} \sqrt {-b e+c (d-e x)}}\right )}{4 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs.
\(2(209)=418\).
time = 0.04, size = 622, normalized size = 2.69
method | result | size |
default | \(-\frac {\left (4 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-7 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}-\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+8 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -14 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +4 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -7 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +4 b \,e^{2} g x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-9 c d e g x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+c \,e^{2} f x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+2 b d e g \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+2 b \,e^{2} f \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-5 c \,d^{2} g \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-3 c d e f \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{4 \left (b e -2 c d \right )^{\frac {3}{2}} e^{2} \sqrt {-c e x -b e +c d}\, \left (e x +d \right )^{\frac {5}{2}}}\) | \(622\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 454 vs.
\(2 (215) = 430\).
time = 2.98, size = 964, normalized size = 4.17 \begin {gather*} \left [\frac {{\left (7 \, c^{2} d^{4} g + {\left (c^{2} f - 4 \, b c g\right )} x^{3} e^{4} + {\left (7 \, c^{2} d g x^{3} + 3 \, {\left (c^{2} d f - 4 \, b c d g\right )} x^{2}\right )} e^{3} + 3 \, {\left (7 \, c^{2} d^{2} g x^{2} + {\left (c^{2} d^{2} f - 4 \, b c d^{2} g\right )} x\right )} e^{2} + {\left (21 \, c^{2} d^{3} g x + c^{2} d^{3} f - 4 \, b c d^{3} g\right )} e\right )} \sqrt {2 \, c d - b e} \log \left (\frac {3 \, c d^{2} - {\left (c x^{2} + 2 \, b x\right )} e^{2} + 2 \, {\left (c d x - b d\right )} e + 2 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {2 \, c d - b e} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (10 \, c^{2} d^{3} g + {\left (2 \, b^{2} f + {\left (b c f + 4 \, b^{2} g\right )} x\right )} e^{3} - {\left (7 \, b c d f - 2 \, b^{2} d g + {\left (2 \, c^{2} d f + 17 \, b c d g\right )} x\right )} e^{2} + 3 \, {\left (6 \, c^{2} d^{2} g x + 2 \, c^{2} d^{2} f - 3 \, b c d^{2} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{8 \, {\left (4 \, c^{2} d^{5} e^{2} + b^{2} x^{3} e^{7} - {\left (4 \, b c d x^{3} - 3 \, b^{2} d x^{2}\right )} e^{6} + {\left (4 \, c^{2} d^{2} x^{3} - 12 \, b c d^{2} x^{2} + 3 \, b^{2} d^{2} x\right )} e^{5} + {\left (12 \, c^{2} d^{3} x^{2} - 12 \, b c d^{3} x + b^{2} d^{3}\right )} e^{4} + 4 \, {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{3}\right )}}, \frac {{\left (7 \, c^{2} d^{4} g + {\left (c^{2} f - 4 \, b c g\right )} x^{3} e^{4} + {\left (7 \, c^{2} d g x^{3} + 3 \, {\left (c^{2} d f - 4 \, b c d g\right )} x^{2}\right )} e^{3} + 3 \, {\left (7 \, c^{2} d^{2} g x^{2} + {\left (c^{2} d^{2} f - 4 \, b c d^{2} g\right )} x\right )} e^{2} + {\left (21 \, c^{2} d^{3} g x + c^{2} d^{3} f - 4 \, b c d^{3} g\right )} e\right )} \sqrt {-2 \, c d + b e} \arctan \left (-\frac {\sqrt {-2 \, c d + b e} \sqrt {x e + d}}{\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}\right ) - {\left (10 \, c^{2} d^{3} g + {\left (2 \, b^{2} f + {\left (b c f + 4 \, b^{2} g\right )} x\right )} e^{3} - {\left (7 \, b c d f - 2 \, b^{2} d g + {\left (2 \, c^{2} d f + 17 \, b c d g\right )} x\right )} e^{2} + 3 \, {\left (6 \, c^{2} d^{2} g x + 2 \, c^{2} d^{2} f - 3 \, b c d^{2} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{4 \, {\left (4 \, c^{2} d^{5} e^{2} + b^{2} x^{3} e^{7} - {\left (4 \, b c d x^{3} - 3 \, b^{2} d x^{2}\right )} e^{6} + {\left (4 \, c^{2} d^{2} x^{3} - 12 \, b c d^{2} x^{2} + 3 \, b^{2} d^{2} x\right )} e^{5} + {\left (12 \, c^{2} d^{3} x^{2} - 12 \, b c d^{3} x + b^{2} d^{3}\right )} e^{4} + 4 \, {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{3}\right )}}\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 {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 354, normalized size = 1.53 \begin {gather*} -\frac {{\left (\frac {{\left (7 \, c^{3} d g + c^{3} f e - 4 \, b c^{2} g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d - b e\right )} \sqrt {-2 \, c d + b e}} + \frac {14 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d^{2} g + 2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d f e - 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} d g e - 9 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} d g - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} f e^{2} + 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{2} g e^{2} + {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} f e + 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{2} g e}{{\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{2} c^{2}}\right )} e^{\left (-2\right )}}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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