Optimal. Leaf size=144 \[ \frac {20 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{21 b^{9/4} d \sqrt {a+b x}}+\frac {20 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b} \]
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Rubi [A] time = 0.08, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 224, 221} \[ \frac {20 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}{21 b^2}+\frac {20 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{21 b^{9/4} d \sqrt {a+b x}}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{\sqrt {a+b x}} \, dx &=\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {(5 (b c-a d)) \int \frac {\sqrt [4]{c+d x}}{\sqrt {a+b x}} \, dx}{7 b}\\ &=\frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{21 b^2}\\ &=\frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {\left (20 (b c-a d)^2\right ) \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 )}{21 b^2 d}\\ &=\frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {\left (20 (b c-a d)^2 \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 )}{21 b^2 d \sqrt {a+b x}}\\ &=\frac {20 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{21 b^2}+\frac {4 \sqrt {a+b x} (c+d x)^{5/4}}{7 b}+\frac {20 (b c-a d)^{9/4} \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 )}{21 b^{9/4} d \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 71, normalized size = 0.49 \[ \frac {2 \sqrt {a+b x} (c+d x)^{5/4} \, _2F_1\left (-\frac {5}{4},\frac {1}{2};\frac {3}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{\frac {5}{4}}}{\sqrt {b x + a}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\sqrt {b x +a}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{\sqrt {b x + a}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^{5/4}}{\sqrt {a+b\,x}} \,d x \]
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 \[ \int \frac {\left (c + d x\right )^{\frac {5}{4}}}{\sqrt {a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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