Optimal. Leaf size=191 \[ 90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-45 c^2 \sqrt [4]{b^2-4 a c} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-45 c^2 \sqrt [4]{b^2-4 a c} d^{11/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {700, 706, 708,
335, 218, 212, 209} \begin {gather*} -45 c^2 d^{11/2} \sqrt [4]{b^2-4 a c} \text {ArcTan}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-45 c^2 d^{11/2} \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}+90 c^2 d^5 \sqrt {b d+2 c d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 700
Rule 706
Rule 708
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (9 c d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (45 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\\ &=90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (45 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{4} \left (45 c \left (b^2-4 a c\right ) d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )\\ &=90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (45 c \left (b^2-4 a c\right ) d^5\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-\left (45 c^2 \sqrt {b^2-4 a c} d^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (45 c^2 \sqrt {b^2-4 a c} d^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-45 c^2 \sqrt [4]{b^2-4 a c} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-45 c^2 \sqrt [4]{b^2-4 a c} d^{11/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.55, size = 298, normalized size = 1.56 \begin {gather*} \left (\frac {1}{2}+\frac {i}{2}\right ) c^2 (d (b+2 c x))^{11/2} \left (\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (45 b^4-360 a b^2 c+720 a^2 c^2-81 b^2 (b+2 c x)^2+324 a c (b+2 c x)^2+32 (b+2 c x)^4\right )}{c^2 (b+2 c x)^5 (a+x (b+c x))^2}-\frac {45 i \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}+\frac {45 i \sqrt [4]{b^2-4 a c} \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}+\frac {45 i \sqrt [4]{b^2-4 a c} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{11/2}}\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. \(358\) vs.
\(2(161)=322\).
time = 0.73, size = 359, normalized size = 1.88
method | result | size |
derivativedivides | \(64 c^{2} d^{5} \left (\sqrt {2 c d x +b d}-d^{2} \left (\frac {16 \left (-\frac {17 a c}{128}+\frac {17 b^{2}}{512}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {13}{32} a^{2} c^{2} d^{2}+\frac {13}{64} a \,b^{2} c \,d^{2}-\frac {13}{512} b^{4} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}+\left (2 c d x +b d \right )^{2}\right )^{2}}+\frac {45 \left (\frac {a c}{4}-\frac {b^{2}}{16}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(359\) |
default | \(64 c^{2} d^{5} \left (\sqrt {2 c d x +b d}-d^{2} \left (\frac {16 \left (-\frac {17 a c}{128}+\frac {17 b^{2}}{512}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {13}{32} a^{2} c^{2} d^{2}+\frac {13}{64} a \,b^{2} c \,d^{2}-\frac {13}{512} b^{4} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}+\left (2 c d x +b d \right )^{2}\right )^{2}}+\frac {45 \left (\frac {a c}{4}-\frac {b^{2}}{16}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(359\) |
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. 523 vs.
\(2 (161) = 322\).
time = 2.75, size = 523, normalized size = 2.74 \begin {gather*} -\frac {180 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\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 )} \arctan \left (\frac {\left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {3}{4}} \sqrt {2 \, c d x + b d} c^{2} d^{5} - \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {3}{4}} \sqrt {2 \, c^{5} d^{11} x + b c^{4} d^{11} + \sqrt {{\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}}}}{{\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}}\right ) + 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\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 )} \log \left (45 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} + 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\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 )} \log \left (45 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - {\left (128 \, c^{4} d^{5} x^{4} + 256 \, b c^{3} d^{5} x^{3} + 3 \, {\left (37 \, b^{2} c^{2} + 108 \, a c^{3}\right )} d^{5} x^{2} - {\left (17 \, b^{3} c - 324 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 9 \, a b^{2} c - 180 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d}}{2 \, {\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 )}} \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. 521 vs.
\(2 (161) = 322\).
time = 2.37, size = 521, normalized size = 2.73 \begin {gather*} -\frac {45}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {45}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {45}{4} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {45}{4} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 64 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} + \frac {2 \, {\left (13 \, \sqrt {2 \, c d x + b d} b^{4} c^{2} d^{9} - 104 \, \sqrt {2 \, c d x + b d} a b^{2} c^{3} d^{9} + 208 \, \sqrt {2 \, c d x + b d} a^{2} c^{4} d^{9} - 17 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} c^{2} d^{7} + 68 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c^{3} d^{7}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 720, normalized size = 3.77 \begin {gather*} \frac {{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (136\,a\,c^3\,d^7-34\,b^2\,c^2\,d^7\right )+\sqrt {b\,d+2\,c\,d\,x}\,\left (416\,a^2\,c^4\,d^9-208\,a\,b^2\,c^3\,d^9+26\,b^4\,c^2\,d^9\right )}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}+64\,c^2\,d^5\,\sqrt {b\,d+2\,c\,d\,x}-45\,c^2\,d^{11/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}-c^2\,d^{11/2}\,\mathrm {atan}\left (\frac {\frac {c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )-\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,45{}\mathrm {i}}{2}+\frac {c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )+\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,45{}\mathrm {i}}{2}}{\frac {45\,c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )-\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}}{2}-\frac {45\,c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )+\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}}{2}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,45{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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