Optimal. Leaf size=289 \[ -\frac {5 e \left (b^2-4 a c\right )^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{8192 c^5}-\frac {5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3072 c^4}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{192 c^3}+\frac {\left (a+b x+c x^2\right )^{7/2} \left (-2 c e (16 a e+9 b d)+9 b^2 e^2+14 c e x (2 c d-b e)+32 c^2 d^2\right )}{504 c^2}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2} \]
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Rubi [A] time = 0.55, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{7/2} \left (-2 c e (16 a e+9 b d)+9 b^2 e^2+14 c e x (2 c d-b e)+32 c^2 d^2\right )}{504 c^2}+\frac {5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{8192 c^5}-\frac {5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3072 c^4}+\frac {e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{192 c^3}-\frac {5 e \left (b^2-4 a c\right )^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\int (d+e x) (2 c (b d-2 a e)+2 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}+\frac {\left (\left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 \left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{384 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}+\frac {\left (5 \left (b^2-4 a c\right )^3 e (2 c d-b e)\right ) \int \sqrt {a+b x+c x^2} \, dx}{2048 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^3 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}-\frac {5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 \left (b^2-4 a c\right )^4 e (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16384 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^3 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}-\frac {5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac {\left (5 \left (b^2-4 a c\right )^4 e (2 c d-b e)\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 )}{8192 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^3 e (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^5}-\frac {5 \left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{3072 c^4}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{192 c^3}+\frac {2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {\left (32 c^2 d^2+9 b^2 e^2-2 c e (9 b d+16 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{504 c^2}-\frac {5 \left (b^2-4 a c\right )^4 e (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 252, normalized size = 0.87 \begin {gather*} -\frac {e \left (b^2-4 a c\right ) (b e-2 c d) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{49152 c^{11/2}}+\frac {(a+x (b+c x))^{7/2} \left (-2 c e (16 a e+9 b d+7 b e x)+9 b^2 e^2+4 c^2 d (8 d+7 e x)\right )}{504 c^2}+\frac {2}{9} (d+e x)^2 (a+x (b+c x))^{7/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 4.86, size = 875, normalized size = 3.03 \begin {gather*} \frac {\sqrt {c x^2+b x+a} \left (-315 e^2 b^8+630 c d e b^7+210 c e^2 x b^7+4620 a c e^2 b^6-168 c^2 e^2 x^2 b^6-420 c^2 d e x b^6+144 c^3 e^2 x^3 b^5+336 c^3 d e x^2 b^5-9240 a c^2 d e b^5-2856 a c^2 e^2 x b^5-128 c^4 e^2 x^4 b^4-288 c^4 d e x^3 b^4-24528 a^2 c^2 e^2 b^4+2112 a c^3 e^2 x^2 b^4+5712 a c^3 d e x b^4+93952 c^5 e^2 x^5 b^3+229632 c^5 d e x^4 b^3+147456 c^5 d^2 x^3 b^3-1664 a c^4 e^2 x^3 b^3-4224 a c^4 d e x^2 b^3+49056 a^2 c^3 d e b^3+13536 a^2 c^3 e^2 x b^3+310272 c^6 e^2 x^6 b^2+729600 c^6 d e x^5 b^2+442368 c^6 d^2 x^4 b^2+241152 a c^5 e^2 x^4 b^2+625920 a c^5 d e x^3 b^2+53568 a^3 c^3 e^2 b^2+442368 a c^5 d^2 x^2 b^2-8832 a^2 c^4 e^2 x^2 b^2-27072 a^2 c^4 d e x b^2+329728 c^7 e^2 x^7 b+755712 c^7 d e x^6 b+442368 c^7 d^2 x^5 b+568320 a c^6 e^2 x^5 b+1385472 a c^6 d e x^4 b+884736 a c^6 d^2 x^3 b+174336 a^2 c^5 e^2 x^3 b+509184 a^2 c^5 d e x^2 b-107136 a^3 c^4 d e b+442368 a^2 c^5 d^2 x b-23936 a^3 c^4 e^2 x b+114688 c^8 e^2 x^8+258048 c^8 d e x^7+147456 c^8 d^2 x^6+311296 a c^7 e^2 x^6+731136 a c^7 d e x^5+442368 a c^7 d^2 x^4+245760 a^2 c^6 e^2 x^4+634368 a^2 c^6 d e x^3+147456 a^3 c^5 d^2-32768 a^4 c^4 e^2+442368 a^2 c^6 d^2 x^2+16384 a^3 c^5 e^2 x^2+80640 a^3 c^5 d e x\right )}{516096 c^5}-\frac {5 \left (e^2 b^9-2 c d e b^8-16 a c e^2 b^7+32 a c^2 d e b^6+96 a^2 c^2 e^2 b^5-192 a^2 c^3 d e b^4-256 a^3 c^3 e^2 b^3+512 a^3 c^4 d e b^2+256 a^4 c^4 e^2 b-512 a^4 c^5 d e\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{16384 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 1523, normalized size = 5.27
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 824, normalized size = 2.85 \begin {gather*} \frac {1}{516096} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (8 \, c^{3} x e^{2} + \frac {18 \, c^{11} d e + 23 \, b c^{10} e^{2}}{c^{8}}\right )} x + \frac {144 \, c^{11} d^{2} + 738 \, b c^{10} d e + 303 \, b^{2} c^{9} e^{2} + 304 \, a c^{10} e^{2}}{c^{8}}\right )} x + \frac {1728 \, b c^{10} d^{2} + 2850 \, b^{2} c^{9} d e + 2856 \, a c^{10} d e + 367 \, b^{3} c^{8} e^{2} + 2220 \, a b c^{9} e^{2}}{c^{8}}\right )} x + \frac {3456 \, b^{2} c^{9} d^{2} + 3456 \, a c^{10} d^{2} + 1794 \, b^{3} c^{8} d e + 10824 \, a b c^{9} d e - b^{4} c^{7} e^{2} + 1884 \, a b^{2} c^{8} e^{2} + 1920 \, a^{2} c^{9} e^{2}}{c^{8}}\right )} x + \frac {9216 \, b^{3} c^{8} d^{2} + 55296 \, a b c^{9} d^{2} - 18 \, b^{4} c^{7} d e + 39120 \, a b^{2} c^{8} d e + 39648 \, a^{2} c^{9} d e + 9 \, b^{5} c^{6} e^{2} - 104 \, a b^{3} c^{7} e^{2} + 10896 \, a^{2} b c^{8} e^{2}}{c^{8}}\right )} x + \frac {55296 \, a b^{2} c^{8} d^{2} + 55296 \, a^{2} c^{9} d^{2} + 42 \, b^{5} c^{6} d e - 528 \, a b^{3} c^{7} d e + 63648 \, a^{2} b c^{8} d e - 21 \, b^{6} c^{5} e^{2} + 264 \, a b^{4} c^{6} e^{2} - 1104 \, a^{2} b^{2} c^{7} e^{2} + 2048 \, a^{3} c^{8} e^{2}}{c^{8}}\right )} x + \frac {221184 \, a^{2} b c^{8} d^{2} - 210 \, b^{6} c^{5} d e + 2856 \, a b^{4} c^{6} d e - 13536 \, a^{2} b^{2} c^{7} d e + 40320 \, a^{3} c^{8} d e + 105 \, b^{7} c^{4} e^{2} - 1428 \, a b^{5} c^{5} e^{2} + 6768 \, a^{2} b^{3} c^{6} e^{2} - 11968 \, a^{3} b c^{7} e^{2}}{c^{8}}\right )} x + \frac {147456 \, a^{3} c^{8} d^{2} + 630 \, b^{7} c^{4} d e - 9240 \, a b^{5} c^{5} d e + 49056 \, a^{2} b^{3} c^{6} d e - 107136 \, a^{3} b c^{7} d e - 315 \, b^{8} c^{3} e^{2} + 4620 \, a b^{6} c^{4} e^{2} - 24528 \, a^{2} b^{4} c^{5} e^{2} + 53568 \, a^{3} b^{2} c^{6} e^{2} - 32768 \, a^{4} c^{7} e^{2}}{c^{8}}\right )} + \frac {5 \, {\left (2 \, b^{8} c d e - 32 \, a b^{6} c^{2} d e + 192 \, a^{2} b^{4} c^{3} d e - 512 \, a^{3} b^{2} c^{4} d e + 512 \, a^{4} c^{5} d e - b^{9} e^{2} + 16 \, a b^{7} c e^{2} - 96 \, a^{2} b^{5} c^{2} e^{2} + 256 \, a^{3} b^{3} c^{3} e^{2} - 256 \, a^{4} b c^{4} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16384 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1358, normalized size = 4.70
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 (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,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 (b + 2 c x\right ) \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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