Optimal. Leaf size=213 \[ \frac {5 d^3 \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{84 b^{9/4} \sqrt {a+b x} (b c-a d)^{7/4}}+\frac {5 d^3 \sqrt [4]{c+d x}}{84 b^2 \sqrt {a+b x} (b c-a d)^2}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (a+b x)^{3/2} (b c-a d)}-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {47, 51, 63, 224, 221} \[ \frac {5 d^3 \sqrt [4]{c+d x}}{84 b^2 \sqrt {a+b x} (b c-a d)^2}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (a+b x)^{3/2} (b c-a d)}+\frac {5 d^3 \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{84 b^{9/4} \sqrt {a+b x} (b c-a d)^{7/4}}-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{(a+b x)^{9/2}} \, dx &=-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}+\frac {(5 d) \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{7/2}} \, dx}{14 b}\\ &=-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}+\frac {d^2 \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/4}} \, dx}{28 b^2}\\ &=-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (b c-a d) (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}-\frac {\left (5 d^3\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx}{168 b^2 (b c-a d)}\\ &=-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {5 d^3 \sqrt [4]{c+d x}}{84 b^2 (b c-a d)^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}+\frac {\left (5 d^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{336 b^2 (b c-a d)^2}\\ &=-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {5 d^3 \sqrt [4]{c+d x}}{84 b^2 (b c-a d)^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}+\frac {\left (5 d^3\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 )}{84 b^2 (b c-a d)^2}\\ &=-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {5 d^3 \sqrt [4]{c+d x}}{84 b^2 (b c-a d)^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}+\frac {\left (5 d^3 \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 )}{84 b^2 (b c-a d)^2 \sqrt {a+b x}}\\ &=-\frac {d \sqrt [4]{c+d x}}{7 b^2 (a+b x)^{5/2}}-\frac {d^2 \sqrt [4]{c+d x}}{42 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {5 d^3 \sqrt [4]{c+d x}}{84 b^2 (b c-a d)^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/4}}{7 b (a+b x)^{7/2}}+\frac {5 d^3 \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 )}{84 b^{9/4} (b c-a d)^{7/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 73, normalized size = 0.34 \[ -\frac {2 (c+d x)^{5/4} \, _2F_1\left (-\frac {7}{2},-\frac {5}{4};-\frac {5}{2};\frac {d (a+b x)}{a d-b c}\right )}{7 b (a+b x)^{7/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{4}}}{b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{\frac {9}{2}}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^{5/4}}{{\left (a+b\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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