Optimal. Leaf size=353 \[ -\frac {2 (a A+(A b+3 a B) x) \sqrt {a+b x+c x^2}}{3 a x^{3/2}}+\frac {2 (A b+6 a B) \sqrt {c} \sqrt {x} \sqrt {a+b x+c x^2}}{3 a \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 (A b+6 a B) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x+c x^2}}+\frac {\left ((A b+6 a B) \sqrt {c}+\sqrt {a} (3 b B+2 A c)\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt [4]{c} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {824, 853, 1211,
1117, 1209} \begin {gather*} \frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {c} (6 a B+A b)+\sqrt {a} (2 A c+3 b B)\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x\right ) (6 a B+A b) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {c} \sqrt {x} (6 a B+A b) \sqrt {a+b x+c x^2}}{3 a \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 824
Rule 853
Rule 1117
Rule 1209
Rule 1211
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx &=-\frac {2 (a A+(A b+3 a B) x) \sqrt {a+b x+c x^2}}{3 a x^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} a (3 b B+2 A c)-\frac {1}{2} (A b+6 a B) c x}{\sqrt {x} \sqrt {a+b x+c x^2}} \, dx}{3 a}\\ &=-\frac {2 (a A+(A b+3 a B) x) \sqrt {a+b x+c x^2}}{3 a x^{3/2}}-\frac {4 \text {Subst}\left (\int \frac {-\frac {1}{2} a (3 b B+2 A c)-\frac {1}{2} (A b+6 a B) c x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a}\\ &=-\frac {2 (a A+(A b+3 a B) x) \sqrt {a+b x+c x^2}}{3 a x^{3/2}}-\frac {\left (2 (A b+6 a B) \sqrt {c}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {a}}+\frac {1}{3} \left (2 \left (3 b B+\frac {(A b+6 a B) \sqrt {c}}{\sqrt {a}}+2 A c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 (a A+(A b+3 a B) x) \sqrt {a+b x+c x^2}}{3 a x^{3/2}}+\frac {2 (A b+6 a B) \sqrt {c} \sqrt {x} \sqrt {a+b x+c x^2}}{3 a \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 (A b+6 a B) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x+c x^2}}+\frac {\left (3 b B+\frac {(A b+6 a B) \sqrt {c}}{\sqrt {a}}+2 A c\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 21.85, size = 499, normalized size = 1.41 \begin {gather*} \frac {-4 (A b x+a (A+3 B x)) (a+x (b+c x))+\frac {x \left (4 (A b+6 a B) \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))+i (A b+6 a B) \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (6 a B \sqrt {b^2-4 a c}+A \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}}{6 a x^{3/2} \sqrt {a+x (b+c x)}} \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. \(1686\) vs.
\(2(345)=690\).
time = 0.95, size = 1687, normalized size = 4.78
method | result | size |
elliptic | \(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 A \sqrt {c \,x^{3}+b \,x^{2}+a x}}{3 x^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right ) \left (A b +3 B a \right )}{3 a \sqrt {x \left (c \,x^{2}+b x +a \right )}}+\frac {\left (\frac {2 A c}{3}+B b \right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {\left (B c +\frac {c \left (A b +3 B a \right )}{3 a}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}\right )}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(791\) |
risch | \(-\frac {2 \sqrt {c \,x^{2}+b x +a}\, \left (A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a}+\frac {\left (\frac {\left (A b c +6 a B c \right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {2 A a \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{\sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {3 a b B \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}\right ) \sqrt {x \left (c \,x^{2}+b x +a \right )}}{3 a \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(1015\) |
default | \(\text {Expression too large to display}\) | \(1687\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.98, size = 198, normalized size = 0.56 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (6 \, B a + A b\right )} c^{\frac {3}{2}} x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - {\left (3 \, B a b - A b^{2} + 6 \, A a c\right )} \sqrt {c} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (A a c + {\left (3 \, B a + A b\right )} c x\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{9 \, a c x^{2}} \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 \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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