Optimal. Leaf size=210 \[ -26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+52 c d^7 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {52}{5} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2} \]
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Rubi [A] time = 0.18, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {686, 692, 694, 329, 212, 206, 203} \begin {gather*} 52 c d^7 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {52}{5} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 686
Rule 692
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c d^2\right ) \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx\\ &=\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\\ &=\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 c \left (b^2-4 a c\right )^3 d^8\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac {1}{2} \left (13 \left (b^2-4 a c\right )^3 d^7\right ) \operatorname {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 )\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\left (13 \left (b^2-4 a c\right )^3 d^7\right ) \operatorname {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 )\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-\left (26 c \left (b^2-4 a c\right )^{5/2} d^8\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (26 c \left (b^2-4 a c\right )^{5/2} d^8\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=52 c \left (b^2-4 a c\right )^2 d^7 \sqrt {b d+2 c d x}+\frac {52}{5} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{5/2}+\frac {52}{9} c d^3 (b d+2 c d x)^{9/2}-\frac {d (b d+2 c d x)^{13/2}}{a+b x+c x^2}-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-26 c \left (b^2-4 a c\right )^{9/4} d^{15/2} \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 [A] time = 0.60, size = 189, normalized size = 0.90 \begin {gather*} -\frac {(d (b+2 c x))^{15/2} \left (-13 \left (b^2-4 a c\right ) \left (3 \left (b^2-4 a c\right ) \left (-30 \left (b^2-4 a c\right ) \sqrt {b+2 c x}-60 c \sqrt [4]{b^2-4 a c} (a+x (b+c x)) \left (\tan ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )+24 (b+2 c x)^{5/2}\right )+8 (b+2 c x)^{9/2}\right )-40 (b+2 c x)^{13/2}\right )}{90 (b+2 c x)^{15/2} (a+x (b+c x))} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.33, size = 424, normalized size = 2.02 \begin {gather*} \frac {\sqrt {b d+2 c d x} \left (37440 a^3 c^3 d^7-20592 a^2 b^2 c^2 d^7+29952 a^2 b c^3 d^7 x+29952 a^2 c^4 d^7 x^2+3068 a b^4 c d^7-16640 a b^3 c^2 d^7 x-19968 a b^2 c^3 d^7 x^2-6656 a b c^4 d^7 x^3-3328 a c^5 d^7 x^4-45 b^6 d^7+2528 b^5 c d^7 x+4320 b^4 c^2 d^7 x^2+4864 b^3 c^3 d^7 x^3+5632 b^2 c^4 d^7 x^4+3840 b c^5 d^7 x^5+1280 c^6 d^7 x^6\right )}{45 \left (a+b x+c x^2\right )}+(13-13 i) c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac {-\frac {(1+i) c \sqrt {d} x}{\sqrt [4]{b^2-4 a c}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {d}}{\sqrt [4]{b^2-4 a c}}+\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {d} \sqrt [4]{b^2-4 a c}}{\sqrt {b d+2 c d x}}\right )-(13-13 i) c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b d+2 c d x}}{\sqrt {d} \left (\sqrt {b^2-4 a c}+i b+2 i c x\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1445, normalized size = 6.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 876, normalized size = 4.17 \begin {gather*} 48 \, \sqrt {2 \, c d x + b d} b^{4} c d^{7} - 384 \, \sqrt {2 \, c d x + b d} a b^{2} c^{2} d^{7} + 768 \, \sqrt {2 \, c d x + b d} a^{2} c^{3} d^{7} + \frac {32}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} c d^{5} - \frac {128}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c^{2} d^{5} + \frac {16}{9} \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c d^{3} - 13 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \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 ) - 13 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \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 {13}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \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 {13}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} c d^{7} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{3} d^{7}\right )} \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 {4 \, {\left (\sqrt {2 \, c d x + b d} b^{6} c d^{9} - 12 \, \sqrt {2 \, c d x + b d} a b^{4} c^{2} d^{9} + 48 \, \sqrt {2 \, c d x + b d} a^{2} b^{2} c^{3} d^{9} - 64 \, \sqrt {2 \, c d x + b d} a^{3} c^{4} d^{9}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1512, normalized size = 7.20
result too large to display
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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mupad [B] time = 0.69, size = 1060, normalized size = 5.05 \begin {gather*} \frac {16\,c\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{9}-\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (-256\,a^3\,c^4\,d^9+192\,a^2\,b^2\,c^3\,d^9-48\,a\,b^4\,c^2\,d^9+4\,b^6\,c\,d^9\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}+48\,c\,d^7\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2-26\,c\,d^{15/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}-\frac {32\,c\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{5}-c\,d^{15/2}\,\mathrm {atan}\left (\frac {c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )\,13{}\mathrm {i}+c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )+13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )\,13{}\mathrm {i}}{13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )-13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (44302336\,a^6\,c^8\,d^{18}-66453504\,a^5\,b^2\,c^7\,d^{18}+41533440\,a^4\,b^4\,c^6\,d^{18}-13844480\,a^3\,b^6\,c^5\,d^{18}+2595840\,a^2\,b^8\,c^4\,d^{18}-259584\,a\,b^{10}\,c^3\,d^{18}+10816\,b^{12}\,c^2\,d^{18}\right )+13\,c\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}\,\left (212992\,a^4\,c^5\,d^{11}-212992\,a^3\,b^2\,c^4\,d^{11}+79872\,a^2\,b^4\,c^3\,d^{11}-13312\,a\,b^6\,c^2\,d^{11}+832\,b^8\,c\,d^{11}\right )\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}\,26{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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