Optimal. Leaf size=193 \[ 77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2} \]
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Rubi [A] time = 0.15, antiderivative size = 193, 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 = {686, 692, 694, 329, 298, 203, 206} \[ 77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 686
Rule 692
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (11 c d^2\right ) \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (77 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx\\ &=\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (77 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx\\ &=\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{4} \left (77 c \left (b^2-4 a c\right ) d^5\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )\\ &=\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (77 c \left (b^2-4 a c\right ) d^5\right ) \operatorname {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}-\left (77 c^2 \left (b^2-4 a c\right ) d^7\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 (77 c^2 \left (b^2-4 a c\right ) d^7\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 )\\ &=\frac {154}{3} c^2 d^5 (b d+2 c d x)^{3/2}-\frac {d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}+77 c^2 \left (b^2-4 a c\right )^{3/4} d^{13/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-77 c^2 \left (b^2-4 a c\right )^{3/4} d^{13/2} \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] time = 0.15, size = 145, normalized size = 0.75 \[ -\frac {4 d^5 (d (b+2 c x))^{3/2} \left (-16 c^2 \left (77 a^2+55 a c x^2+5 c^2 x^4\right )+1232 c^2 (a+x (b+c x))^2 \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )+4 b^2 c \left (99 a+25 c x^2\right )-80 b c^2 x \left (11 a+2 c x^2\right )-27 b^4+180 b^3 c x\right )}{15 (a+x (b+c x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 951, normalized size = 4.93 \[ \frac {924 \, \left ({\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\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^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\right )^{\frac {1}{4}} {\left (b^{4} c^{6} - 8 \, a b^{2} c^{7} + 16 \, a^{2} c^{8}\right )} \sqrt {2 \, c d x + b d} d^{19} - \sqrt {2 \, {\left (b^{8} c^{13} - 16 \, a b^{6} c^{14} + 96 \, a^{2} b^{4} c^{15} - 256 \, a^{3} b^{2} c^{16} + 256 \, a^{4} c^{17}\right )} d^{39} x + {\left (b^{9} c^{12} - 16 \, a b^{7} c^{13} + 96 \, a^{2} b^{5} c^{14} - 256 \, a^{3} b^{3} c^{15} + 256 \, a^{4} b c^{16}\right )} d^{39} + \sqrt {{\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}} {\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}} \left ({\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\right )^{\frac {1}{4}}}{{\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}}\right ) - 231 \, \left ({\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\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 (456533 \, {\left (b^{4} c^{6} - 8 \, a b^{2} c^{7} + 16 \, a^{2} c^{8}\right )} \sqrt {2 \, c d x + b d} d^{19} + 456533 \, \left ({\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\right )^{\frac {3}{4}}\right ) + 231 \, \left ({\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\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 (456533 \, {\left (b^{4} c^{6} - 8 \, a b^{2} c^{7} + 16 \, a^{2} c^{8}\right )} \sqrt {2 \, c d x + b d} d^{19} - 456533 \, \left ({\left (b^{6} c^{8} - 12 \, a b^{4} c^{9} + 48 \, a^{2} b^{2} c^{10} - 64 \, a^{3} c^{11}\right )} d^{26}\right )^{\frac {3}{4}}\right ) + {\left (256 \, c^{5} d^{6} x^{5} + 640 \, b c^{4} d^{6} x^{4} + 2 \, {\left (199 \, b^{2} c^{3} + 484 \, a c^{4}\right )} d^{6} x^{3} - {\left (43 \, b^{3} c^{2} - 1452 \, a b c^{3}\right )} d^{6} x^{2} - {\left (63 \, b^{4} c - 418 \, a b^{2} c^{2} - 616 \, a^{2} c^{3}\right )} d^{6} x - {\left (3 \, b^{5} + 33 \, a b^{3} c - 308 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt {2 \, c d x + b d}}{6 \, {\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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 521, normalized size = 2.70 \[ -\frac {77}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {77}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) + \frac {77}{4} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{5} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac {77}{4} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{5} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {64}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c^{2} d^{5} + \frac {2 \, {\left (15 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} c^{2} d^{9} - 120 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a b^{2} c^{3} d^{9} + 240 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a^{2} c^{4} d^{9} - 19 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} c^{2} d^{7} + 76 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} a c^{3} d^{7}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 857, normalized size = 4.44 \[ \frac {480 \left (2 c d x +b d \right )^{\frac {3}{2}} a^{2} c^{4} d^{9}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}-\frac {240 \left (2 c d x +b d \right )^{\frac {3}{2}} a \,b^{2} c^{3} d^{9}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {30 \left (2 c d x +b d \right )^{\frac {3}{2}} b^{4} c^{2} d^{9}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {154 \sqrt {2}\, a \,c^{3} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}-\frac {154 \sqrt {2}\, a \,c^{3} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}-\frac {77 \sqrt {2}\, a \,c^{3} d^{7} \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 )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}-\frac {77 \sqrt {2}\, b^{2} c^{2} d^{7} \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 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+\frac {77 \sqrt {2}\, b^{2} c^{2} d^{7} \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 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+\frac {77 \sqrt {2}\, b^{2} c^{2} d^{7} \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 )}{4 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+\frac {152 \left (2 c d x +b d \right )^{\frac {7}{2}} a \,c^{3} d^{7}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}-\frac {38 \left (2 c d x +b d \right )^{\frac {7}{2}} b^{2} c^{2} d^{7}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {64 \left (2 c d x +b d \right )^{\frac {3}{2}} c^{2} d^{5}}{3} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 264, normalized size = 1.37 \[ \frac {{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\left (152\,a\,c^3\,d^7-38\,b^2\,c^2\,d^7\right )+{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (480\,a^2\,c^4\,d^9-240\,a\,b^2\,c^3\,d^9+30\,b^4\,c^2\,d^9\right )}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}+\frac {64\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}{3}+77\,c^2\,d^{13/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}+c^2\,d^{13/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}\,77{}\mathrm {i} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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