Optimal. Leaf size=135 \[ -\frac {128 c (b+2 c x) (2 c d-b e)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {16 (b+2 c x) (2 c d-b e)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {638, 614, 613} \begin {gather*} -\frac {128 c (b+2 c x) (2 c d-b e)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {16 (b+2 c x) (2 c d-b e)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rule 638
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {(8 (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 \left (b^2-4 a c\right )}\\ &=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {16 (2 c d-b e) (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {(64 c (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {16 (2 c d-b e) (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 119, normalized size = 0.88 \begin {gather*} \frac {2 \left (3 \left (b^2-4 a c\right )^2 (2 a e-b d+b e x-2 c d x)-8 \left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x)) (b e-2 c d)+64 c (b+2 c x) (a+x (b+c x))^2 (b e-2 c d)\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.18, size = 267, normalized size = 1.98 \begin {gather*} -\frac {2 \left (-96 a^3 c^2 e-48 a^2 b^2 c e+240 a^2 b c^2 d-240 a^2 b c^2 e x+480 a^2 c^3 d x+2 a b^4 e-40 a b^3 c d-120 a b^3 c e x+240 a b^2 c^2 d x-480 a b^2 c^2 e x^2+960 a b c^3 d x^2-320 a b c^3 e x^3+640 a c^4 d x^3+3 b^5 d+5 b^5 e x-10 b^4 c d x-40 b^4 c e x^2+80 b^3 c^2 d x^2-240 b^3 c^2 e x^3+480 b^2 c^3 d x^3-320 b^2 c^3 e x^4+640 b c^4 d x^4-128 b c^4 e x^5+256 c^5 d x^5\right )}{15 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.31, size = 550, normalized size = 4.07 \begin {gather*} -\frac {2 \, {\left (128 \, {\left (2 \, c^{5} d - b c^{4} e\right )} x^{5} + 320 \, {\left (2 \, b c^{4} d - b^{2} c^{3} e\right )} x^{4} + 80 \, {\left (2 \, {\left (3 \, b^{2} c^{3} + 4 \, a c^{4}\right )} d - {\left (3 \, b^{3} c^{2} + 4 \, a b c^{3}\right )} e\right )} x^{3} + 40 \, {\left (2 \, {\left (b^{3} c^{2} + 12 \, a b c^{3}\right )} d - {\left (b^{4} c + 12 \, a b^{2} c^{2}\right )} e\right )} x^{2} + {\left (3 \, b^{5} - 40 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} d + 2 \, {\left (a b^{4} - 24 \, a^{2} b^{2} c - 48 \, a^{3} c^{2}\right )} e - 5 \, {\left (2 \, {\left (b^{4} c - 24 \, a b^{2} c^{2} - 48 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 24 \, a b^{3} c - 48 \, a^{2} b c^{2}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15 \, {\left (a^{3} b^{6} - 12 \, a^{4} b^{4} c + 48 \, a^{5} b^{2} c^{2} - 64 \, a^{6} c^{3} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{6} + 3 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{5} + 3 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} x^{4} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} x^{3} + 3 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 448, normalized size = 3.32 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (2 \, c^{5} d - b c^{4} e\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (2 \, b c^{4} d - b^{2} c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (6 \, b^{2} c^{3} d + 8 \, a c^{4} d - 3 \, b^{3} c^{2} e - 4 \, a b c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (2 \, b^{3} c^{2} d + 24 \, a b c^{3} d - b^{4} c e - 12 \, a b^{2} c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (2 \, b^{4} c d - 48 \, a b^{2} c^{2} d - 96 \, a^{2} c^{3} d - b^{5} e + 24 \, a b^{3} c e + 48 \, a^{2} b c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {3 \, b^{5} d - 40 \, a b^{3} c d + 240 \, a^{2} b c^{2} d + 2 \, a b^{4} e - 48 \, a^{2} b^{2} c e - 96 \, a^{3} c^{2} e}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 288, normalized size = 2.13 \begin {gather*} -\frac {2 \left (128 b \,c^{4} e \,x^{5}-256 c^{5} d \,x^{5}+320 b^{2} c^{3} e \,x^{4}-640 b \,c^{4} d \,x^{4}+320 a b \,c^{3} e \,x^{3}-640 a \,c^{4} d \,x^{3}+240 b^{3} c^{2} e \,x^{3}-480 b^{2} c^{3} d \,x^{3}+480 a \,b^{2} c^{2} e \,x^{2}-960 a b \,c^{3} d \,x^{2}+40 b^{4} c e \,x^{2}-80 b^{3} c^{2} d \,x^{2}+240 a^{2} b \,c^{2} e x -480 a^{2} c^{3} d x +120 a \,b^{3} c e x -240 a \,b^{2} c^{2} d x -5 b^{5} e x +10 b^{4} c d x +96 a^{3} c^{2} e +48 a^{2} b^{2} c e -240 a^{2} b \,c^{2} d -2 a \,b^{4} e +40 a \,b^{3} c d -3 b^{5} d \right )}{15 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )} \end {gather*}
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 [B] time = 1.95, size = 395, normalized size = 2.93 \begin {gather*} \frac {x\,\left (\frac {4\,c^2\,d}{5\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {2\,b\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )-\frac {4\,a\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,b\,c\,d}{5\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}-\frac {x\,\left (\frac {2\,c^2\,\left (20\,b\,e-32\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,b\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {b\,c\,\left (20\,b\,e-32\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {16\,a\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {\frac {b\,c\,\left (256\,c^2\,d-128\,b\,c\,e\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {2\,c^2\,x\,\left (256\,c^2\,d-128\,b\,c\,e\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {c\,x^2+b\,x+a}}-\frac {4\,e}{\left (60\,a\,c-15\,b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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