Optimal. Leaf size=350 \[ \frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {5 \sqrt {c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]
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Rubi [A]
time = 0.36, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {806, 676, 678,
635, 210} \begin {gather*} \frac {5 \sqrt {c} (2 c d-b e) \text {ArcTan}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (3 b e g-10 c d g+4 c e f)}{8 e^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 635
Rule 676
Rule 678
Rule 806
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 )^{5/2}}{(d+e x)^5} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}-\frac {(4 c e f-10 c d g+3 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx}{3 e (2 c d-b e)}\\ &=\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (4 c e f-10 c d g+3 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)^2} \, dx}{3 e (2 c d-b e)}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (4 c e f-10 c d g+3 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{4 e}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \text {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{4 e}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {5 \sqrt {c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2}\\ \end {align*}
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Mathematica [A]
time = 2.15, size = 267, normalized size = 0.76 \begin {gather*} \frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (-\sqrt {-b e+c (d-e x)} \left (b c e \left (-147 d^2 g+d e (24 f-206 g x)+e^2 x (56 f-27 g x)\right )+8 b^2 e^2 (2 d g+e (f+3 g x))+2 c^2 \left (118 d^3 g-23 d^2 e (2 f-7 g x)-3 e^3 x^2 (2 f+g x)+4 d e^2 x (-17 f+6 g x)\right )\right )+15 \sqrt {-c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) (d+e x)^{3/2} \log \left (-\sqrt {-c} \sqrt {d+e x}+\sqrt {c d-b e-c e x}\right )\right )}{12 e^2 (d+e x)^4 (-b e+c (d-e x))^{5/2}} \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. \(1254\) vs.
\(2(324)=648\).
time = 0.05, size = 1255, normalized size = 3.59
method | result | size |
default | \(\text {Expression too large to display}\) | \(1255\) |
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 = 9.00, size = 854, normalized size = 2.44 \begin {gather*} \left [\frac {15 \, {\left (20 \, c^{2} d^{4} g + {\left (4 \, b c f + 3 \, b^{2} g\right )} x^{2} e^{4} - 2 \, {\left (4 \, {\left (c^{2} d f + 2 \, b c d g\right )} x^{2} - {\left (4 \, b c d f + 3 \, b^{2} d g\right )} x\right )} e^{3} + {\left (20 \, c^{2} d^{2} g x^{2} + 4 \, b c d^{2} f + 3 \, b^{2} d^{2} g - 16 \, {\left (c^{2} d^{2} f + 2 \, b c d^{2} g\right )} x\right )} e^{2} + 8 \, {\left (5 \, c^{2} d^{3} g x - c^{2} d^{3} f - 2 \, b c d^{3} g\right )} e\right )} \sqrt {-c} \log \left (-4 \, c^{2} d^{2} + 4 \, b c d e - 4 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {-c} e + {\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2}\right )} e^{2}\right ) - 4 \, {\left (236 \, c^{2} d^{3} g - {\left (6 \, c^{2} g x^{3} - 8 \, b^{2} f + 3 \, {\left (4 \, c^{2} f + 9 \, b c g\right )} x^{2} - 8 \, {\left (7 \, b c f + 3 \, b^{2} g\right )} x\right )} e^{3} + 2 \, {\left (24 \, c^{2} d g x^{2} + 12 \, b c d f + 8 \, b^{2} d g - {\left (68 \, c^{2} d f + 103 \, b c d g\right )} x\right )} e^{2} + {\left (322 \, c^{2} d^{2} g x - 92 \, c^{2} d^{2} f - 147 \, b c d^{2} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}{48 \, {\left (x^{2} e^{4} + 2 \, d x e^{3} + d^{2} e^{2}\right )}}, \frac {15 \, {\left (20 \, c^{2} d^{4} g + {\left (4 \, b c f + 3 \, b^{2} g\right )} x^{2} e^{4} - 2 \, {\left (4 \, {\left (c^{2} d f + 2 \, b c d g\right )} x^{2} - {\left (4 \, b c d f + 3 \, b^{2} d g\right )} x\right )} e^{3} + {\left (20 \, c^{2} d^{2} g x^{2} + 4 \, b c d^{2} f + 3 \, b^{2} d^{2} g - 16 \, {\left (c^{2} d^{2} f + 2 \, b c d^{2} g\right )} x\right )} e^{2} + 8 \, {\left (5 \, c^{2} d^{3} g x - c^{2} d^{3} f - 2 \, b c d^{3} g\right )} e\right )} \sqrt {c} \arctan \left (-\frac {\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {c} e}{2 \, {\left (c^{2} d^{2} - b c d e - {\left (c^{2} x^{2} + b c x\right )} e^{2}\right )}}\right ) - 2 \, {\left (236 \, c^{2} d^{3} g - {\left (6 \, c^{2} g x^{3} - 8 \, b^{2} f + 3 \, {\left (4 \, c^{2} f + 9 \, b c g\right )} x^{2} - 8 \, {\left (7 \, b c f + 3 \, b^{2} g\right )} x\right )} e^{3} + 2 \, {\left (24 \, c^{2} d g x^{2} + 12 \, b c d f + 8 \, b^{2} d g - {\left (68 \, c^{2} d f + 103 \, b c d g\right )} x\right )} e^{2} + {\left (322 \, c^{2} d^{2} g x - 92 \, c^{2} d^{2} f - 147 \, b c d^{2} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}{24 \, {\left (x^{2} e^{4} + 2 \, d x e^{3} + d^{2} e^{2}\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 {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, 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. 1103 vs.
\(2 (331) = 662\).
time = 1.35, size = 1103, normalized size = 3.15 \begin {gather*} \frac {1}{12} \, {\left (\frac {15 \, {\left (20 \, c^{3} d^{2} g \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 8 \, c^{3} d f e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 16 \, b c^{2} d g e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 4 \, b c^{2} f e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{2} c g e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}}}{\sqrt {c}}\right ) e^{\left (-3\right )}}{\sqrt {c}} - 8 \, {\left (24 \, c^{2} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d^{2} g e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, c {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} d^{2} g e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 12 \, c^{2} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d f e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 2 \, c {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} d f e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 18 \, b c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d g e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + b {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} d g e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 6 \, b c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} f e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} f e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{2} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} g e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-9\right )} - \frac {3 \, {\left (36 \, c^{4} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d^{2} g \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 44 \, c^{3} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} d^{2} g \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 8 \, c^{4} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d f e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 8 \, c^{3} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} d f e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 32 \, b c^{3} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} d g e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 40 \, b c^{2} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} d g e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 4 \, b c^{3} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} f e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 4 \, b c^{2} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} f e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 7 \, b^{2} c^{2} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} g e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 9 \, b^{2} c {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}} g e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-3\right )}}{{\left (\frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{2}}\right )} e \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 )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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