Optimal. Leaf size=674 \[ -\frac {d^2 \left (\frac {c}{d}+x\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{8 a \left (4 a d^2+c^3\right )^{3/2} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{8 a c \left (4 a d^2+c^3\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\left (-c^{3/2} \sqrt {4 a d^2+c^3}+4 a d^2+c^3\right ) \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{16 a c^{5/4} d \left (4 a d^2+c^3\right )^{3/4} \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\sqrt [4]{c} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) E\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{8 a d \sqrt [4]{4 a d^2+c^3} \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]
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Rubi [A] time = 0.68, antiderivative size = 674, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1106, 1092, 1197, 1103, 1195} \[ -\frac {d^2 \left (\frac {c}{d}+x\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{8 a \left (4 a d^2+c^3\right )^{3/2} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{8 a c \left (4 a d^2+c^3\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\left (-c^{3/2} \sqrt {4 a d^2+c^3}+4 a d^2+c^3\right ) \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{16 a c^{5/4} d \left (4 a d^2+c^3\right )^{3/4} \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\sqrt [4]{c} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) E\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{8 a d \sqrt [4]{4 a d^2+c^3} \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1092
Rule 1103
Rule 1106
Rule 1195
Rule 1197
Rubi steps
\begin {align*} \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4\right )^{3/2}} \, dx,x,\frac {c}{d}+x\right )\\ &=-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{8 a c \left (c^3+4 a d^2\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\operatorname {Subst}\left (\int \frac {2 c \left (4 a+\frac {c^3}{d^2}\right ) d^2-2 c^2 d^2 x^2}{\sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac {c}{d}+x\right )}{16 a c^2 \left (c^3+4 a d^2\right )}\\ &=-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{8 a c \left (c^3+4 a d^2\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1-\frac {d^2 x^2}{\sqrt {c} \sqrt {c^3+4 a d^2}}}{\sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac {c}{d}+x\right )}{8 a \sqrt {c^3+4 a d^2}}+\frac {\left (c^3+4 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac {c}{d}+x\right )}{8 a c \left (c^3+4 a d^2\right )}\\ &=-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{8 a c \left (c^3+4 a d^2\right ) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}-\frac {d (c+d x) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{8 a \left (c^3+4 a d^2\right )^{3/2} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )}+\frac {\sqrt [4]{c} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right ) E\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{8 a d \sqrt [4]{c^3+4 a d^2} \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {\left (c^3+4 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right ) F\left (2 \tan ^{-1}\left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{16 a c^{5/4} d \left (c^3+4 a d^2\right )^{3/4} \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 6.13, size = 5276, normalized size = 7.83 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}}{d^{4} x^{8} + 8 \, c d^{3} x^{7} + 24 \, c^{2} d^{2} x^{6} + 32 \, c^{3} d x^{5} + 32 \, a c^{2} d x^{3} + 32 \, a c^{3} x^{2} + 8 \, {\left (2 \, c^{4} + a c d^{2}\right )} x^{4} + 16 \, a^{2} c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 5024, normalized size = 7.45 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (4\,c^2\,x^2+4\,c\,d\,x^3+4\,a\,c+d^2\,x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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