Optimal. Leaf size=96 \[ -\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+16 c^{3/2} d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {700, 635, 212}
\begin {gather*} 16 c^{3/2} d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 700
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}+\left (4 c d^2\right ) \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (16 c^2 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+\left (32 c^2 d^4\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )\\ &=-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+16 c^{3/2} d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 84, normalized size = 0.88 \begin {gather*} d^4 \left (-\frac {2 (b+2 c x) \left (b^2+16 b c x+4 c \left (3 a+4 c x^2\right )\right )}{3 (a+x (b+c x))^{3/2}}-16 c^{3/2} \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right ) \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. \(1136\) vs.
\(2(82)=164\).
time = 0.68, size = 1137, normalized size = 11.84
method | result | size |
default | \(\text {Expression too large to display}\) | \(1137\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (82) = 164\).
time = 7.33, size = 440, normalized size = 4.58 \begin {gather*} \left [\frac {2 \, {\left (12 \, {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\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 ) - {\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, -\frac {2 \, {\left (24 \, {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\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 (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 315 vs.
\(2 (82) = 164\).
time = 2.02, size = 315, normalized size = 3.28 \begin {gather*} -16 \, c^{\frac {3}{2}} d^{4} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right ) - \frac {2 \, {\left (2 \, {\left (8 \, {\left (\frac {2 \, {\left (b^{4} c^{3} d^{4} - 8 \, a b^{2} c^{4} d^{4} + 16 \, a^{2} c^{5} d^{4}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (b^{5} c^{2} d^{4} - 8 \, a b^{3} c^{3} d^{4} + 16 \, a^{2} b c^{4} d^{4}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (3 \, b^{6} c d^{4} - 20 \, a b^{4} c^{2} d^{4} + 16 \, a^{2} b^{2} c^{3} d^{4} + 64 \, a^{3} c^{4} d^{4}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{7} d^{4} + 4 \, a b^{5} c d^{4} - 80 \, a^{2} b^{3} c^{2} d^{4} + 192 \, a^{3} b c^{3} d^{4}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^4}{{\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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