Optimal. Leaf size=290 \[ -\frac {c^{3/4} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{15/4}}-\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{15/4}}+\frac {2 x^{3/2} (b c-a d)^2}{3 d^3}-\frac {2 b x^{7/2} (b c-2 a d)}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d} \]
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Rubi [A] time = 0.25, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {461, 321, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {c^{3/4} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{15/4}}-\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{15/4}}-\frac {2 b x^{7/2} (b c-2 a d)}{7 d^2}+\frac {2 x^{3/2} (b c-a d)^2}{3 d^3}+\frac {2 b^2 x^{11/2}}{11 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 321
Rule 329
Rule 461
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {b (b c-2 a d) x^{5/2}}{d^2}+\frac {b^2 x^{9/2}}{d}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^{5/2}}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}+\frac {(b c-a d)^2 \int \frac {x^{5/2}}{c+d x^2} \, dx}{d^2}\\ &=\frac {2 (b c-a d)^2 x^{3/2}}{3 d^3}-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}-\frac {\left (c (b c-a d)^2\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{d^3}\\ &=\frac {2 (b c-a d)^2 x^{3/2}}{3 d^3}-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}-\frac {\left (2 c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^3}\\ &=\frac {2 (b c-a d)^2 x^{3/2}}{3 d^3}-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}+\frac {\left (c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^{7/2}}-\frac {\left (c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^{7/2}}\\ &=\frac {2 (b c-a d)^2 x^{3/2}}{3 d^3}-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}-\frac {\left (c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 d^4}-\frac {\left (c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 d^4}-\frac {\left (c^{3/4} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} d^{15/4}}-\frac {\left (c^{3/4} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} d^{15/4}}\\ &=\frac {2 (b c-a d)^2 x^{3/2}}{3 d^3}-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}-\frac {c^{3/4} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}-\frac {\left (c^{3/4} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{15/4}}+\frac {\left (c^{3/4} (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{15/4}}\\ &=\frac {2 (b c-a d)^2 x^{3/2}}{3 d^3}-\frac {2 b (b c-2 a d) x^{7/2}}{7 d^2}+\frac {2 b^2 x^{11/2}}{11 d}+\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{15/4}}-\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{15/4}}-\frac {c^{3/4} (b c-a d)^2 \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{15/4}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 276, normalized size = 0.95 \begin {gather*} \frac {-231 \sqrt {2} c^{3/4} (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+231 \sqrt {2} c^{3/4} (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )+462 \sqrt {2} c^{3/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )-462 \sqrt {2} c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )-264 b d^{7/4} x^{7/2} (b c-2 a d)+616 d^{3/4} x^{3/2} (b c-a d)^2+168 b^2 d^{11/4} x^{11/2}}{924 d^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 198, normalized size = 0.68 \begin {gather*} \frac {2 x^{3/2} \left (77 a^2 d^2-154 a b c d+66 a b d^2 x^2+77 b^2 c^2-33 b^2 c d x^2+21 b^2 d^2 x^4\right )}{231 d^3}+\frac {c^{3/4} (b c-a d)^2 \tan ^{-1}\left (\frac {\frac {\sqrt [4]{c}}{\sqrt {2} \sqrt [4]{d}}-\frac {\sqrt [4]{d} x}{\sqrt {2} \sqrt [4]{c}}}{\sqrt {x}}\right )}{\sqrt {2} d^{15/4}}+\frac {c^{3/4} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} d^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.21, size = 1701, normalized size = 5.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 385, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{6}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{2 \, d^{6}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{6}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{4 \, d^{6}} + \frac {2 \, {\left (21 \, b^{2} d^{10} x^{\frac {11}{2}} - 33 \, b^{2} c d^{9} x^{\frac {7}{2}} + 66 \, a b d^{10} x^{\frac {7}{2}} + 77 \, b^{2} c^{2} d^{8} x^{\frac {3}{2}} - 154 \, a b c d^{9} x^{\frac {3}{2}} + 77 \, a^{2} d^{10} x^{\frac {3}{2}}\right )}}{231 \, d^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 504, normalized size = 1.74 \begin {gather*} \frac {2 b^{2} x^{\frac {11}{2}}}{11 d}+\frac {4 a b \,x^{\frac {7}{2}}}{7 d}-\frac {2 b^{2} c \,x^{\frac {7}{2}}}{7 d^{2}}+\frac {2 a^{2} x^{\frac {3}{2}}}{3 d}-\frac {4 a b c \,x^{\frac {3}{2}}}{3 d^{2}}+\frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3 d^{3}}-\frac {\sqrt {2}\, a^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{2}}-\frac {\sqrt {2}\, a^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{2}}-\frac {\sqrt {2}\, a^{2} c \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{2}}+\frac {\sqrt {2}\, a b \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{\left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}+\frac {\sqrt {2}\, a b \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{\left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}+\frac {\sqrt {2}\, a b \,c^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{2 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}-\frac {\sqrt {2}\, b^{2} c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}}-\frac {\sqrt {2}\, b^{2} c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}}-\frac {\sqrt {2}\, b^{2} c^{3} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.45, size = 263, normalized size = 0.91 \begin {gather*} -\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, d^{3}} + \frac {2 \, {\left (21 \, b^{2} d^{2} x^{\frac {11}{2}} - 33 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{\frac {7}{2}} + 77 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}}\right )}}{231 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 435, normalized size = 1.50 \begin {gather*} x^{3/2}\,\left (\frac {2\,a^2}{3\,d}+\frac {c\,\left (\frac {2\,b^2\,c}{d^2}-\frac {4\,a\,b}{d}\right )}{3\,d}\right )-x^{7/2}\,\left (\frac {2\,b^2\,c}{7\,d^2}-\frac {4\,a\,b}{7\,d}\right )+\frac {2\,b^2\,x^{11/2}}{11\,d}-\frac {{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{3/4}\,d^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a^4\,c^3\,d^4-4\,a^3\,b\,c^4\,d^3+6\,a^2\,b^2\,c^5\,d^2-4\,a\,b^3\,c^6\,d+b^4\,c^7\right )}{a^6\,c^4\,d^6-6\,a^5\,b\,c^5\,d^5+15\,a^4\,b^2\,c^6\,d^4-20\,a^3\,b^3\,c^7\,d^3+15\,a^2\,b^4\,c^8\,d^2-6\,a\,b^5\,c^9\,d+b^6\,c^{10}}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{15/4}}-\frac {{\left (-c\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{3/4}\,d^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a^4\,c^3\,d^4-4\,a^3\,b\,c^4\,d^3+6\,a^2\,b^2\,c^5\,d^2-4\,a\,b^3\,c^6\,d+b^4\,c^7\right )\,1{}\mathrm {i}}{a^6\,c^4\,d^6-6\,a^5\,b\,c^5\,d^5+15\,a^4\,b^2\,c^6\,d^4-20\,a^3\,b^3\,c^7\,d^3+15\,a^2\,b^4\,c^8\,d^2-6\,a\,b^5\,c^9\,d+b^6\,c^{10}}\right )\,{\left (a\,d-b\,c\right )}^2\,1{}\mathrm {i}}{d^{15/4}} \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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