Optimal. Leaf size=159 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}+\frac {4}{d^3 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 159, 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 = {693, 694, 329, 298, 203, 206} \begin {gather*} \frac {4}{d^3 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 693
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )} \, dx &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^4}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\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 )}{2 c \left (b^2-4 a c\right )^2 d^5}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {\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 )}{c \left (b^2-4 a c\right )^2 d^5}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}-\frac {2 \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 (b^2-4 a c\right )^2 d^3}+\frac {2 \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 (b^2-4 a c\right )^2 d^3}\\ &=\frac {4}{5 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2}}+\frac {4}{\left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 56, normalized size = 0.35 \begin {gather*} \frac {4 \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{5 d \left (b^2-4 a c\right ) (d (b+2 c x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.47, size = 255, normalized size = 1.60 \begin {gather*} \frac {8 \left (-2 a c+3 b^2+10 b c x+10 c^2 x^2\right ) \sqrt {b d+2 c d x}}{5 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3}-\frac {(1+i) \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 )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac {(1+i) \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 )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1693, normalized size = 10.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 605, normalized size = 3.81 \begin {gather*} -\frac {\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} - \frac {\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} + \frac {{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{\sqrt {2} b^{6} d^{5} - 12 \, \sqrt {2} a b^{4} c d^{5} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt {2} a^{3} c^{3} d^{5}} - \frac {{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} \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 )}{\sqrt {2} b^{6} d^{5} - 12 \, \sqrt {2} a b^{4} c d^{5} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt {2} a^{3} c^{3} d^{5}} + \frac {4 \, {\left (b^{2} d^{2} - 4 \, a c d^{2} + 5 \, {\left (2 \, c d x + b d\right )}^{2}\right )}}{5 \, {\left (b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}\right )} {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 369, normalized size = 2.32 \begin {gather*} -\frac {4}{5 \left (4 a c -b^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}} d}-\frac {\sqrt {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 )}{\left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d^{3}}+\frac {\sqrt {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 )}{\left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d^{3}}+\frac {\sqrt {2}\, \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 \left (4 a c -b^{2}\right )^{2} \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} d^{3}}+\frac {4}{\left (4 a c -b^{2}\right )^{2} \sqrt {2 c d x +b d}\, d^{3}} \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 = 0.26, size = 230, normalized size = 1.45 \begin {gather*} \frac {\frac {4}{5\,\left (b^2\,d-4\,a\,c\,d\right )}+\frac {4\,{\left (b\,d+2\,c\,d\,x\right )}^2}{d\,{\left (b^2\,d-4\,a\,c\,d\right )}^2}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}+\frac {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 )}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}}+\frac {\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,2{}\mathrm {i}}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}} \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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