Optimal. Leaf size=111 \[ \frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {8 (b c-a d)^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 \sqrt [4]{b} d^2 \sqrt {a+b x}} \]
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Rubi [A] time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 224, 221} \[ \frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {8 (b c-a d)^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt [4]{b} d^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx &=\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {(2 (b c-a d)) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{3 d}\\ &=\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {(8 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 d^2}\\ &=\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {\left (8 (b c-a d) \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {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 )}{3 d^2 \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d}-\frac {8 (b c-a d)^{5/4} \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 )}{3 \sqrt [4]{b} d^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.66 \[ \frac {2 (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {5}{2};\frac {d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x +a}}{\left (d x +c \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{3/4}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x}}{\left (c + d x\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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