Optimal. Leaf size=232 \[ \frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d^2 \sqrt {a+b x} \sqrt [4]{b c-a d}}-\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt {a+b x} \sqrt [4]{b c-a d}}+\frac {8 b \sqrt {a+b x}}{5 d \sqrt [4]{c+d x} (b c-a d)}-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}} \]
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Rubi [A] time = 0.23, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {47, 51, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt {a+b x} \sqrt [4]{b c-a d}}-\frac {8 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^2 \sqrt {a+b x} \sqrt [4]{b c-a d}}+\frac {8 b \sqrt {a+b x}}{5 d \sqrt [4]{c+d x} (b c-a d)}-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 221
Rule 224
Rule 307
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{(c+d x)^{9/4}} \, dx &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {(2 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx}{5 d}\\ &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {\left (2 b^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{5 d (b c-a d)}\\ &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {\left (8 b^2\right ) \operatorname {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 )}{5 d^2 (b c-a d)}\\ &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}+\frac {\left (8 b^{3/2}\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 )}{5 d^2 \sqrt {b c-a d}}-\frac {\left (8 b^{3/2}\right ) \operatorname {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 )}{5 d^2 \sqrt {b c-a d}}\\ &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}+\frac {\left (8 b^{3/2} \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 )}{5 d^2 \sqrt {b c-a d} \sqrt {a+b x}}-\frac {\left (8 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {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 )}{5 d^2 \sqrt {b c-a d} \sqrt {a+b x}}\\ &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}+\frac {8 b^{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 )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}}-\frac {\left (8 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {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 )}{5 d^2 \sqrt {b c-a d} \sqrt {a+b x}}\\ &=-\frac {4 \sqrt {a+b x}}{5 d (c+d x)^{5/4}}+\frac {8 b \sqrt {a+b x}}{5 d (b c-a d) \sqrt [4]{c+d x}}-\frac {8 b^{5/4} \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 )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}}+\frac {8 b^{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 )}{5 d^2 \sqrt [4]{b c-a d} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 73, normalized size = 0.31 \[ \frac {2 (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \, _2F_1\left (\frac {3}{2},\frac {9}{4};\frac {5}{2};\frac {d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{9/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\right ) \]
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 \[ \int \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x +a}}{\left (d x +c \right )^{\frac {9}{4}}}\, dx \]
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 \[ \int \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{9/4}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x}}{\left (c + d x\right )^{\frac {9}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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