Optimal. Leaf size=268 \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}+\frac {a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (-\frac {b \sqrt {c}}{\sqrt {d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}+\frac {a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (\frac {b \sqrt {c}}{\sqrt {d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )} \]
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Rubi [A] time = 0.27, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {747, 401, 108, 409, 1218, 444, 63, 212, 208, 205} \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}+\frac {a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (-\frac {b \sqrt {c}}{\sqrt {d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}+\frac {a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (\frac {b \sqrt {c}}{\sqrt {d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 108
Rule 205
Rule 208
Rule 212
Rule 401
Rule 409
Rule 444
Rule 747
Rule 1218
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \left (c+d x^2\right )^{3/4}} \, dx &=a \int \frac {1}{\left (a^2-b^2 x^2\right ) \left (c+d x^2\right )^{3/4}} \, dx-b \int \frac {x}{\left (a^2-b^2 x^2\right ) \left (c+d x^2\right )^{3/4}} \, dx\\ &=-\left (\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{\left (a^2-b^2 x\right ) (c+d x)^{3/4}} \, dx,x,x^2\right )\right )+\frac {\left (a \sqrt {-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-\frac {d x}{c}} \left (a^2-b^2 x\right ) (c+d x)^{3/4}} \, dx,x,x^2\right )}{2 x}\\ &=-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{a^2+\frac {b^2 c}{d}-\frac {b^2 x^4}{d}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{d}-\frac {\left (2 a \sqrt {-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-b^2 c-a^2 d+b^2 x^4\right ) \sqrt {1-\frac {x^4}{c}}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{x}\\ &=-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2 c+a^2 d}-b x^2} \, dx,x,\sqrt [4]{c+d x^2}\right )}{\sqrt {b^2 c+a^2 d}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2 c+a^2 d}+b x^2} \, dx,x,\sqrt [4]{c+d x^2}\right )}{\sqrt {b^2 c+a^2 d}}+\frac {\left (a \sqrt {-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{\sqrt {b^2 c+a^2 d}}\right ) \sqrt {1-\frac {x^4}{c}}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{\left (b^2 c+a^2 d\right ) x}+\frac {\left (a \sqrt {-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{\sqrt {b^2 c+a^2 d}}\right ) \sqrt {1-\frac {x^4}{c}}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{\left (b^2 c+a^2 d\right ) x}\\ &=-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{b^2 c+a^2 d}}\right )}{\left (b^2 c+a^2 d\right )^{3/4}}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{b^2 c+a^2 d}}\right )}{\left (b^2 c+a^2 d\right )^{3/4}}+\frac {a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (-\frac {b \sqrt {c}}{\sqrt {b^2 c+a^2 d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c+d x^2}}{\sqrt [4]{c}}\right )\right |-1\right )}{\left (b^2 c+a^2 d\right ) x}+\frac {a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (\frac {b \sqrt {c}}{\sqrt {b^2 c+a^2 d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c+d x^2}}{\sqrt [4]{c}}\right )\right |-1\right )}{\left (b^2 c+a^2 d\right ) x}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 233, normalized size = 0.87 \[ \frac {-\sqrt {b} x \sqrt [4]{a^2 d+b^2 c} \left (\tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )\right )+a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (-\frac {b \sqrt {c}}{\sqrt {d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )+a \sqrt [4]{c} \sqrt {-\frac {d x^2}{c}} \Pi \left (\frac {b \sqrt {c}}{\sqrt {d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x^{2} + c\right )}^{\frac {3}{4}} {\left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right ) \left (d \,x^{2}+c \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (d x^{2} + c\right )}^{\frac {3}{4}} {\left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d\,x^2+c\right )}^{3/4}\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right ) \left (c + d x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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