Optimal. Leaf size=182 \[ -\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{14 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac {\left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{14 c^3 \sqrt {d} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {699, 705, 703,
227} \begin {gather*} \frac {\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 )}{14 c^3 \sqrt {d} \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{14 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 699
Rule 703
Rule 705
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx &=\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}-\frac {\left (3 \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}{14 c}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{14 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{28 c^2}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{14 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac {\left (\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}{28 c^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{14 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac {\left (\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 )}{14 c^3 d \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{14 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac {\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 )}{14 c^3 \sqrt {d} \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 = 0.54 \begin {gather*} -\frac {\left (b^2-4 a c\right ) \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \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. \(565\) vs.
\(2(154)=308\).
time = 0.78, size = 566, normalized size = 3.11 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 = 0.65, size = 119, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (2 \, c^{4} x^{2} + 2 \, b c^{3} x - b^{2} c^{2} + 6 \, a c^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{28 \, c^{4} d} \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 {3}{2}}}{\sqrt {d \left (b + 2 c x\right )}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{\sqrt {b\,d+2\,c\,d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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