Optimal. Leaf size=219 \[ -\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^3}+\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{84 c^4 d^{5/2} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {698, 699, 705,
703, 227} \begin {gather*} \frac {5 \left (b^2-4 a c\right )^{9/4} \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 )}{84 c^4 d^{5/2} \sqrt {a+b x+c x^2}}-\frac {5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{84 c^3 d^3}+\frac {5 \left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 698
Rule 699
Rule 703
Rule 705
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{5/2}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx}{6 c d^2}\\ &=\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx}{28 c^2 d^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^3}+\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{168 c^3 d^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^3}+\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {\left (5 \left (b^2-4 a c\right )^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\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}{168 c^3 d^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^3}+\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {\left (5 \left (b^2-4 a c\right )^2 \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 )}{84 c^4 d^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^3}+\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {5 \left (b^2-4 a c\right )^{9/4} \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 )}{84 c^4 d^{5/2} \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 = 10.06, size = 101, normalized size = 0.46 \begin {gather*} -\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {5}{2},-\frac {3}{4};\frac {1}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{96 c^3 d (d (b+2 c x))^{3/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \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. \(999\) vs.
\(2(185)=370\).
time = 0.82, size = 1000, normalized size = 4.57 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 = 0.54, size = 259, normalized size = 1.18 \begin {gather*} \frac {5 \, \sqrt {2} {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 4 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (12 \, c^{6} x^{4} + 24 \, b c^{5} x^{3} - 5 \, b^{4} c^{2} + 30 \, a b^{2} c^{3} - 28 \, a^{2} c^{4} + 2 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 32 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{168 \, {\left (4 \, c^{7} d^{3} x^{2} + 4 \, b c^{6} d^{3} x + b^{2} c^{5} d^{3}\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 {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d \left (b + 2 c x\right )\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 {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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