Optimal. Leaf size=321 \[ -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 793, 626,
635, 212} \begin {gather*} \frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}+\frac {e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 756
Rule 793
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (\frac {1}{2} \left (14 c d^2-e (5 b d+4 a e)\right )+\frac {9}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{16 c^3}\\ &=\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^4}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 2.07, size = 506, normalized size = 1.58 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c e \left (-90 a e^2+c \left (90 d^2+35 d e x+6 e^2 x^2\right )\right )+16 b^2 c^2 \left (343 a^2 e^3-2 a c e \left (525 d^2+189 d e x+31 e^2 x^2\right )+2 c^2 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )\right )-16 b^3 c^2 \left (-7 a e^2 (95 d+13 e x)+c \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )\right )-32 b c^3 \left (a^2 e^2 (567 d+73 e x)-2 a c \left (175 d^3+147 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )-4 c^2 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )\right )+64 c^3 \left (-32 a^3 e^3+a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )+4 c^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+2 a c^2 x \left (175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right )\right )\right )+105 \left (b^2-4 a c\right )^2 (-2 c d+b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{71680 c^{11/2}} \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. \(933\) vs.
\(2(295)=590\).
time = 0.82, size = 934, normalized size = 2.91 Too large to display
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 661 vs.
\(2 (307) = 614\).
time = 2.37, size = 1325, normalized size = 4.13 \begin {gather*} \left [\frac {105 \, {\left (16 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} - 24 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{2} e + 2 \, {\left (7 \, b^{6} c - 60 \, a b^{4} c^{2} + 144 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d e^{2} - {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (8960 \, c^{7} d^{3} x^{3} + 13440 \, b c^{6} d^{3} x^{2} + 1120 \, {\left (b^{2} c^{5} + 20 \, a c^{6}\right )} d^{3} x - 560 \, {\left (3 \, b^{3} c^{4} - 20 \, a b c^{5}\right )} d^{3} + {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} e^{3} + 14 \, {\left (1280 \, c^{7} d x^{5} + 1664 \, b c^{6} d x^{4} + 16 \, {\left (3 \, b^{2} c^{5} + 140 \, a c^{6}\right )} d x^{3} - 8 \, {\left (7 \, b^{3} c^{4} - 36 \, a b c^{5}\right )} d x^{2} + 2 \, {\left (35 \, b^{4} c^{3} - 216 \, a b^{2} c^{4} + 240 \, a^{2} c^{5}\right )} d x - {\left (105 \, b^{5} c^{2} - 760 \, a b^{3} c^{3} + 1296 \, a^{2} b c^{4}\right )} d\right )} e^{2} + 168 \, {\left (128 \, c^{7} d^{2} x^{4} + 176 \, b c^{6} d^{2} x^{3} + 8 \, {\left (b^{2} c^{5} + 32 \, a c^{6}\right )} d^{2} x^{2} - 2 \, {\left (5 \, b^{3} c^{4} - 28 \, a b c^{5}\right )} d^{2} x + {\left (15 \, b^{4} c^{3} - 100 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{143360 \, c^{6}}, -\frac {105 \, {\left (16 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} - 24 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{2} e + 2 \, {\left (7 \, b^{6} c - 60 \, a b^{4} c^{2} + 144 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d e^{2} - {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (8960 \, c^{7} d^{3} x^{3} + 13440 \, b c^{6} d^{3} x^{2} + 1120 \, {\left (b^{2} c^{5} + 20 \, a c^{6}\right )} d^{3} x - 560 \, {\left (3 \, b^{3} c^{4} - 20 \, a b c^{5}\right )} d^{3} + {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} e^{3} + 14 \, {\left (1280 \, c^{7} d x^{5} + 1664 \, b c^{6} d x^{4} + 16 \, {\left (3 \, b^{2} c^{5} + 140 \, a c^{6}\right )} d x^{3} - 8 \, {\left (7 \, b^{3} c^{4} - 36 \, a b c^{5}\right )} d x^{2} + 2 \, {\left (35 \, b^{4} c^{3} - 216 \, a b^{2} c^{4} + 240 \, a^{2} c^{5}\right )} d x - {\left (105 \, b^{5} c^{2} - 760 \, a b^{3} c^{3} + 1296 \, a^{2} b c^{4}\right )} d\right )} e^{2} + 168 \, {\left (128 \, c^{7} d^{2} x^{4} + 176 \, b c^{6} d^{2} x^{3} + 8 \, {\left (b^{2} c^{5} + 32 \, a c^{6}\right )} d^{2} x^{2} - 2 \, {\left (5 \, b^{3} c^{4} - 28 \, a b c^{5}\right )} d^{2} x + {\left (15 \, b^{4} c^{3} - 100 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{71680 \, c^{6}}\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 \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, 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. 715 vs.
\(2 (307) = 614\).
time = 2.27, size = 715, normalized size = 2.23 \begin {gather*} \frac {1}{35840} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c x e^{3} + \frac {14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac {168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3} + 64 \, a c^{6} e^{3}}{c^{6}}\right )} x + \frac {560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} + 1960 \, a c^{6} d e^{2} - 9 \, b^{3} c^{4} e^{3} + 44 \, a b c^{5} e^{3}}{c^{6}}\right )} x + \frac {1680 \, b c^{6} d^{3} + 168 \, b^{2} c^{5} d^{2} e + 5376 \, a c^{6} d^{2} e - 98 \, b^{3} c^{4} d e^{2} + 504 \, a b c^{5} d e^{2} + 21 \, b^{4} c^{3} e^{3} - 124 \, a b^{2} c^{4} e^{3} + 128 \, a^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac {560 \, b^{2} c^{5} d^{3} + 11200 \, a c^{6} d^{3} - 840 \, b^{3} c^{4} d^{2} e + 4704 \, a b c^{5} d^{2} e + 490 \, b^{4} c^{3} d e^{2} - 3024 \, a b^{2} c^{4} d e^{2} + 3360 \, a^{2} c^{5} d e^{2} - 105 \, b^{5} c^{2} e^{3} + 728 \, a b^{3} c^{3} e^{3} - 1168 \, a^{2} b c^{4} e^{3}}{c^{6}}\right )} x - \frac {1680 \, b^{3} c^{4} d^{3} - 11200 \, a b c^{5} d^{3} - 2520 \, b^{4} c^{3} d^{2} e + 16800 \, a b^{2} c^{4} d^{2} e - 21504 \, a^{2} c^{5} d^{2} e + 1470 \, b^{5} c^{2} d e^{2} - 10640 \, a b^{3} c^{3} d e^{2} + 18144 \, a^{2} b c^{4} d e^{2} - 315 \, b^{6} c e^{3} + 2520 \, a b^{4} c^{2} e^{3} - 5488 \, a^{2} b^{2} c^{3} e^{3} + 2048 \, a^{3} c^{4} e^{3}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, b^{4} c^{3} d^{3} - 128 \, a b^{2} c^{4} d^{3} + 256 \, a^{2} c^{5} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 192 \, a b^{3} c^{3} d^{2} e - 384 \, a^{2} b c^{4} d^{2} e + 14 \, b^{6} c d e^{2} - 120 \, a b^{4} c^{2} d e^{2} + 288 \, a^{2} b^{2} c^{3} d e^{2} - 128 \, a^{3} c^{4} d e^{2} - 3 \, b^{7} e^{3} + 28 \, a b^{5} c e^{3} - 80 \, a^{2} b^{3} c^{2} e^{3} + 64 \, a^{3} b c^{3} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \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 {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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