Optimal. Leaf size=197 \[ \frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt [4]{b c-a d} \sqrt {a+b x}}+\frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt [4]{b c-a d} \sqrt {a+b x}} \]
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Rubi [A]
time = 0.15, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {53, 65, 313,
230, 227, 1214, 1213, 435} \begin {gather*} \frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt {a+b x} \sqrt [4]{b c-a d}}-\frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt {a+b x} \sqrt [4]{b c-a d}}+\frac {4 \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 227
Rule 230
Rule 313
Rule 435
Rule 1213
Rule 1214
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx &=\frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac {b \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{b c-a d}\\ &=\frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac {(4 b) \text {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d (b c-a d)}\\ &=\frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}+\frac {\left (4 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt {b c-a d}}-\frac {\left (4 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt {b c-a d}}\\ &=\frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}+\frac {\left (4 \sqrt {b} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt {b c-a d} \sqrt {a+b x}}-\frac {\left (4 \sqrt {b} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt {b c-a d} \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}+\frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt [4]{b c-a d} \sqrt {a+b x}}-\frac {\left (4 \sqrt {b} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d \sqrt {b c-a d} \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x}}{(b c-a d) \sqrt [4]{c+d x}}-\frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt [4]{b c-a d} \sqrt {a+b x}}+\frac {4 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d \sqrt [4]{b c-a d} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 71, normalized size = 0.36 \begin {gather*} \frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{4}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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