Optimal. Leaf size=321 \[ \frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}+\frac {e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]
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Rubi [A] time = 0.35, 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 = {742, 779, 612, 621, 206} \begin {gather*} \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 {(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 {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 {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 {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 206
Rule 612
Rule 621
Rule 742
Rule 779
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 ) \operatorname {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 = 0.48, size = 231, normalized size = 0.72 \begin {gather*} \frac {\frac {e (a+x (b+c x))^{5/2} \left (-2 c e (8 a e+49 b d+15 b e x)+21 b^2 e^2+4 c^2 d (32 d+15 e x)\right )}{40 c^2}+\frac {7 (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 ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2048 c^{9/2}}+e (d+e x)^2 (a+x (b+c x))^{5/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 3.17, size = 764, normalized size = 2.38 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-2048 a^3 c^3 e^3+5488 a^2 b^2 c^2 e^3-18144 a^2 b c^3 d e^2-2336 a^2 b c^3 e^3 x+21504 a^2 c^4 d^2 e+6720 a^2 c^4 d e^2 x+1024 a^2 c^4 e^3 x^2-2520 a b^4 c e^3+10640 a b^3 c^2 d e^2+1456 a b^3 c^2 e^3 x-16800 a b^2 c^3 d^2 e-6048 a b^2 c^3 d e^2 x-992 a b^2 c^3 e^3 x^2+11200 a b c^4 d^3+9408 a b c^4 d^2 e x+4032 a b c^4 d e^2 x^2+704 a b c^4 e^3 x^3+22400 a c^5 d^3 x+43008 a c^5 d^2 e x^2+31360 a c^5 d e^2 x^3+8192 a c^5 e^3 x^4+315 b^6 e^3-1470 b^5 c d e^2-210 b^5 c e^3 x+2520 b^4 c^2 d^2 e+980 b^4 c^2 d e^2 x+168 b^4 c^2 e^3 x^2-1680 b^3 c^3 d^3-1680 b^3 c^3 d^2 e x-784 b^3 c^3 d e^2 x^2-144 b^3 c^3 e^3 x^3+1120 b^2 c^4 d^3 x+1344 b^2 c^4 d^2 e x^2+672 b^2 c^4 d e^2 x^3+128 b^2 c^4 e^3 x^4+13440 b c^5 d^3 x^2+29568 b c^5 d^2 e x^3+23296 b c^5 d e^2 x^4+6400 b c^5 e^3 x^5+8960 c^6 d^3 x^3+21504 c^6 d^2 e x^4+17920 c^6 d e^2 x^5+5120 c^6 e^3 x^6\right )}{35840 c^5}+\frac {3 \left (-64 a^3 b c^3 e^3+128 a^3 c^4 d e^2+80 a^2 b^3 c^2 e^3-288 a^2 b^2 c^3 d e^2+384 a^2 b c^4 d^2 e-256 a^2 c^5 d^3-28 a b^5 c e^3+120 a b^4 c^2 d e^2-192 a b^3 c^3 d^2 e+128 a b^2 c^4 d^3+3 b^7 e^3-14 b^6 c d e^2+24 b^5 c^2 d^2 e-16 b^4 c^3 d^3\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 1359, normalized size = 4.23
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, 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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maple [B] time = 0.07, size = 1437, normalized size = 4.48
result 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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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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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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