Optimal. Leaf size=220 \[ -\frac {8 (b c-a d)^3 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac {4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {16 (b c-a d)^{17/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^3 \sqrt {a+b x}} \]
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Rubi [A]
time = 0.12, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {52, 65, 230,
227} \begin {gather*} \frac {16 (b c-a d)^{17/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^3 \sqrt {a+b x}}-\frac {8 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{231 b^2 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac {4 (a+b x)^{5/2} \sqrt [4]{c+d x} (b c-a d)}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 227
Rule 230
Rubi steps
\begin {align*} \int (a+b x)^{3/2} (c+d x)^{5/4} \, dx &=\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {(b c-a d) \int (a+b x)^{3/2} \sqrt [4]{c+d x} \, dx}{3 b}\\ &=\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {(b c-a d)^2 \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx}{33 b^2}\\ &=\frac {4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}-\frac {\left (2 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx}{77 b^2 d}\\ &=-\frac {8 (b c-a d)^3 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac {4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {\left (4 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{231 b^2 d^2}\\ &=-\frac {8 (b c-a d)^3 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac {4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {\left (16 (b c-a d)^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{231 b^2 d^3}\\ &=-\frac {8 (b c-a d)^3 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac {4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {\left (16 (b c-a d)^4 \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {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 )}{231 b^2 d^3 \sqrt {a+b x}}\\ &=-\frac {8 (b c-a d)^3 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac {4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac {4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac {16 (b c-a d)^{17/4} \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 )}{231 b^{9/4} d^3 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.07, size = 73, normalized size = 0.33 \begin {gather*} \frac {2 (a+b x)^{5/2} (c+d x)^{5/4} \, _2F_1\left (-\frac {5}{4},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{-b c+a d}\right )}{5 b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{4}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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