Optimal. Leaf size=158 \[ \frac {4 e^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {4 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {768, 738, 640, 621, 206} \begin {gather*} \frac {4 e^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {4 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 738
Rule 768
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}+(2 e) \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {(4 e) \int \frac {-e (b d-2 a e)-e (2 c d-b e) x}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e^2 (2 c d-b e) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left (2 e^3\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c}\\ &=-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e^2 (2 c d-b e) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left (4 e^3\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 )}{c}\\ &=-\frac {2 (d+e x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {4 e^2 (2 c d-b e) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {2 e^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 285, normalized size = 1.80 \begin {gather*} \frac {6 e^3 \sqrt {a+x (b+c x)} \left (4 a^2 c+a \left (-b^2+4 b c x+4 c^2 x^2\right )-b^2 x (b+c x)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (6 b e \left (a^2 e^2+a c \left (d^2-6 d e x-e^2 x^2\right )+c^2 d x^2 (3 d-2 e x)\right )-4 c \left (3 a^2 e^2 (2 d+e x)+a c \left (d^3+9 d e^2 x^2+4 e^3 x^3\right )-3 c^2 d^2 e x^3\right )+b^2 \left (12 a e^3 x+c \left (d^3+9 d^2 e x-9 d e^2 x^2+7 e^3 x^3\right )\right )+6 b^3 e^3 x^2\right )}{3 c^{3/2} \left (4 a c-b^2\right ) (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.67, size = 273, normalized size = 1.73 \begin {gather*} \frac {2 \left (6 a^2 b e^3-24 a^2 c d e^2-12 a^2 c e^3 x+12 a b^2 e^3 x+6 a b c d^2 e-36 a b c d e^2 x-6 a b c e^3 x^2-4 a c^2 d^3-36 a c^2 d e^2 x^2-16 a c^2 e^3 x^3+6 b^3 e^3 x^2+b^2 c d^3+9 b^2 c d^2 e x-9 b^2 c d e^2 x^2+7 b^2 c e^3 x^3+18 b c^2 d^2 e x^2-12 b c^2 d e^2 x^3+12 c^3 d^2 e x^3\right )}{3 c \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e^3 \log \left (-2 c^{3/2} \sqrt {a+b x+c x^2}+b c+2 c^2 x\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.41, size = 958, normalized size = 6.06 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{4} + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} e^{3} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{3} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\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 ) - 2 \, {\left (6 \, a b c^{2} d^{2} e - 24 \, a^{2} c^{2} d e^{2} + 6 \, a^{2} b c e^{3} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} + {\left (12 \, c^{4} d^{2} e - 12 \, b c^{3} d e^{2} + {\left (7 \, b^{2} c^{2} - 16 \, a c^{3}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, b c^{3} d^{2} e - 3 \, {\left (b^{2} c^{2} + 4 \, a c^{3}\right )} d e^{2} + 2 \, {\left (b^{3} c - a b c^{2}\right )} e^{3}\right )} x^{2} + 3 \, {\left (3 \, b^{2} c^{2} d^{2} e - 12 \, a b c^{2} d e^{2} + 4 \, {\left (a b^{2} c - a^{2} c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{4} + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} e^{3} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} e^{3} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\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 ) + {\left (6 \, a b c^{2} d^{2} e - 24 \, a^{2} c^{2} d e^{2} + 6 \, a^{2} b c e^{3} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} + {\left (12 \, c^{4} d^{2} e - 12 \, b c^{3} d e^{2} + {\left (7 \, b^{2} c^{2} - 16 \, a c^{3}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, b c^{3} d^{2} e - 3 \, {\left (b^{2} c^{2} + 4 \, a c^{3}\right )} d e^{2} + 2 \, {\left (b^{3} c - a b c^{2}\right )} e^{3}\right )} x^{2} + 3 \, {\left (3 \, b^{2} c^{2} d^{2} e - 12 \, a b c^{2} d e^{2} + 4 \, {\left (a b^{2} c - a^{2} c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 492, normalized size = 3.11 \begin {gather*} -\frac {2 \, e^{3} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left ({\left ({\left (\frac {{\left (12 \, b^{2} c^{3} d^{2} e - 48 \, a c^{4} d^{2} e - 12 \, b^{3} c^{2} d e^{2} + 48 \, a b c^{3} d e^{2} + 7 \, b^{4} c e^{3} - 44 \, a b^{2} c^{2} e^{3} + 64 \, a^{2} c^{3} e^{3}\right )} x}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}} + \frac {3 \, {\left (6 \, b^{3} c^{2} d^{2} e - 24 \, a b c^{3} d^{2} e - 3 \, b^{4} c d e^{2} + 48 \, a^{2} c^{3} d e^{2} + 2 \, b^{5} e^{3} - 10 \, a b^{3} c e^{3} + 8 \, a^{2} b c^{2} e^{3}\right )}}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}}\right )} x + \frac {3 \, {\left (3 \, b^{4} c d^{2} e - 12 \, a b^{2} c^{2} d^{2} e - 12 \, a b^{3} c d e^{2} + 48 \, a^{2} b c^{2} d e^{2} + 4 \, a b^{4} e^{3} - 20 \, a^{2} b^{2} c e^{3} + 16 \, a^{3} c^{2} e^{3}\right )}}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}}\right )} x + \frac {b^{4} c d^{3} - 8 \, a b^{2} c^{2} d^{3} + 16 \, a^{2} c^{3} d^{3} + 6 \, a b^{3} c d^{2} e - 24 \, a^{2} b c^{2} d^{2} e - 24 \, a^{2} b^{2} c d e^{2} + 96 \, a^{3} c^{2} d e^{2} + 6 \, a^{2} b^{3} e^{3} - 24 \, a^{3} b c e^{3}}{b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 865, normalized size = 5.47 \begin {gather*} -\frac {32 a b c d \,e^{2} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {32 a \,c^{2} d^{2} e x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {8 b^{3} d \,e^{2} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {8 b^{2} c \,d^{2} e x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {16 a \,b^{2} d \,e^{2}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {16 a b c \,d^{2} e}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {4 a b d \,e^{2} x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {4 a c \,d^{2} e x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {4 b^{4} d \,e^{2}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c}+\frac {b^{3} d \,e^{2} x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}-\frac {4 b^{3} d^{2} e}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {b^{2} d^{2} e x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {2 e^{3} x^{3}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {2 a \,b^{2} d \,e^{2}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}+\frac {2 a b \,d^{2} e}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b^{4} d \,e^{2}}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}-\frac {b^{3} d^{2} e}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}+\frac {2 b^{2} e^{3} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {6 d \,e^{2} x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b^{3} e^{3}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 b d \,e^{2} x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}-\frac {3 d^{2} e x}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {4 a d \,e^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}+\frac {b^{2} d \,e^{2}}{2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}-\frac {b \,d^{2} e}{2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}-\frac {2 e^{3} x}{\sqrt {c \,x^{2}+b x +a}\, c}-\frac {2 d^{3}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {2 e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}+\frac {b \,e^{3}}{\sqrt {c \,x^{2}+b x +a}\, c^{2}} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^3}{{\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(-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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