Optimal. Leaf size=278 \[ -\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}-\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}+\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {701, 705, 704,
313, 227, 1213, 435} \begin {gather*} \frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}-\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}+\frac {4 c (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 313
Rule 435
Rule 701
Rule 704
Rule 705
Rule 1213
Rubi steps
\begin {align*} \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac {(2 c) \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}-\frac {\left (4 c^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}-\frac {\left (4 c^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}-\frac {\left (8 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (8 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {a+b x+c x^2}}-\frac {\left (8 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}-\frac {\left (8 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}-\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}+\frac {8 c \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 99, normalized size = 0.36 \begin {gather*} \frac {32 c (d (b+2 c x))^{3/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \, _2F_1\left (\frac {3}{4},\frac {5}{2};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{3 \left (b^2-4 a c\right )^2 d \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. \(865\) vs.
\(2(240)=480\).
time = 0.78, size = 866, normalized size = 3.12 Too large to display
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 = 0.34, size = 300, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (12 \, \sqrt {2} {\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c + {\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} - b^{3} + 10 \, a b c + 4 \, {\left (b^{2} c + 5 \, a c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \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 {\sqrt {d \left (b + 2 c x\right )}}{\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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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 {\sqrt {b\,d+2\,c\,d\,x}}{{\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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