Optimal. Leaf size=170 \[ \frac {21 c^2 d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac {21 c^2 d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {686, 694, 329, 298, 203, 206} \begin {gather*} \frac {21 c^2 d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac {21 c^2 d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 686
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (7 c d^2\right ) \int \frac {(b d+2 c d x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (21 c^2 d^4\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx\\ &=-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{4} \left (21 c d^3\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 {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (21 c d^3\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 {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}-\left (21 c^2 d^5\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 (21 c^2 d^5\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 {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}+\frac {21 c^2 d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac {21 c^2 d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\sqrt [4]{b^2-4 a c}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 119, normalized size = 0.70 \begin {gather*} \frac {4 (d (b+2 c x))^{9/2} \left (-112 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 )-5 \left (b^2-4 a c\right ) (b+2 c x)^2+7 \left (b^2-4 a c\right )^2\right )}{5 \left (b^2-4 a c\right ) (b+2 c x)^3 (a+x (b+c x))^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.98, size = 297, normalized size = 1.75 \begin {gather*} -\frac {\left (\frac {21}{2}+\frac {21 i}{2}\right ) c^2 d^{9/2} \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 )}{\sqrt [4]{b^2-4 a c}}-\frac {\left (\frac {21}{2}+\frac {21 i}{2}\right ) c^2 d^{9/2} \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 )}{\sqrt [4]{b^2-4 a c}}+\frac {\sqrt {b d+2 c d x} \left (-7 a b c d^4-14 a c^2 d^4 x-b^3 d^4-13 b^2 c d^4 x-33 b c^2 d^4 x^2-22 c^3 d^4 x^3\right )}{2 \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 500, normalized size = 2.94 \begin {gather*} -\frac {84 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\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 (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {1}{4}} \sqrt {2 \, c d x + b d} c^{6} d^{13} - \sqrt {2 \, c^{13} d^{27} x + b c^{12} d^{27} + \sqrt {\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}} {\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{18}} \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {1}{4}}}{c^{8} d^{18}}\right ) + 21 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\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 (9261 \, \sqrt {2 \, c d x + b d} c^{6} d^{13} + 9261 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {3}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) - 21 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\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 (9261 \, \sqrt {2 \, c d x + b d} c^{6} d^{13} - 9261 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {3}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) + {\left (22 \, c^{3} d^{4} x^{3} + 33 \, b c^{2} d^{4} x^{2} + {\left (13 \, b^{2} c + 14 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 7 \, a b c\right )} d^{4}\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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giac [B] time = 0.32, size = 510, normalized size = 3.00 \begin {gather*} -\frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \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 )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} - \frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \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 )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} + \frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \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 )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} - \frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \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 )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} + \frac {2 \, {\left (7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{7} - 28 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{3} d^{7} - 11 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2} d^{5}\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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maple [B] time = 0.07, size = 435, normalized size = 2.56 \begin {gather*} -\frac {56 \left (2 c d x +b d \right )^{\frac {3}{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 {14 \left (2 c d x +b d \right )^{\frac {3}{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 {21 \sqrt {2}\, c^{2} d^{5} \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 {21 \sqrt {2}\, c^{2} d^{5} \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 {21 \sqrt {2}\, c^{2} d^{5} \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 {22 \left (2 c d x +b d \right )^{\frac {7}{2}} c^{2} d^{5}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}} \end {gather*}
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 = 0.59, size = 215, normalized size = 1.26 \begin {gather*} \frac {21\,c^2\,d^{9/2}\,\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}}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (56\,a\,c^3\,d^7-14\,b^2\,c^2\,d^7\right )+22\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{{\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 {21\,c^2\,d^{9/2}\,\mathrm {atanh}\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}} \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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