Optimal. Leaf size=145 \[ -\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 b^{5/4} \sqrt {a+b x} (b c-a d)^{3/4}}-\frac {d \sqrt [4]{c+d x}}{3 b \sqrt {a+b x} (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 b^{5/4} \sqrt {a+b x} (b c-a d)^{3/4}}-\frac {d \sqrt [4]{c+d x}}{3 b \sqrt {a+b x} (b c-a d)}-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{5/2}} \, dx &=-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}}+\frac {d \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx}{6 b}\\ &=-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {d \sqrt [4]{c+d x}}{3 b (b c-a d) \sqrt {a+b x}}-\frac {d^2 \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{12 b (b c-a d)}\\ &=-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {d \sqrt [4]{c+d x}}{3 b (b c-a d) \sqrt {a+b x}}-\frac {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 b (b c-a d)}\\ &=-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {d \sqrt [4]{c+d x}}{3 b (b c-a d) \sqrt {a+b x}}-\frac {\left (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 b (b c-a d) \sqrt {a+b x}}\\ &=-\frac {2 \sqrt [4]{c+d x}}{3 b (a+b x)^{3/2}}-\frac {d \sqrt [4]{c+d x}}{3 b (b c-a d) \sqrt {a+b x}}-\frac {d \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 b^{5/4} (b c-a d)^{3/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 73, normalized size = 0.50 \[ -\frac {2 \sqrt [4]{c+d x} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};-\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{4}}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\left (d x +c \right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {1}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^{1/4}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sqrt [4]{c + d x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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