Optimal. Leaf size=182 \[ \frac {20 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac {40 (b c-a d)^{13/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^2 \sqrt {a+b x}} \]
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Rubi [A]
time = 0.07, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {40 (b c-a d)^{13/4} \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^2 \sqrt {a+b x}}+\frac {20 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac {20 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 227
Rule 230
Rubi steps
\begin {align*} \int \sqrt {a+b x} (c+d x)^{5/4} \, dx &=\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}+\frac {(5 (b c-a d)) \int \sqrt {a+b x} \sqrt [4]{c+d x} \, dx}{11 b}\\ &=\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx}{77 b^2}\\ &=\frac {20 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac {\left (10 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{231 b^2 d}\\ &=\frac {20 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac {\left (40 (b c-a d)^3\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^2}\\ &=\frac {20 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac {\left (40 (b c-a d)^3 \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^2 \sqrt {a+b x}}\\ &=\frac {20 (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac {20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac {40 (b c-a d)^{13/4} \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^2 \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 = 10.04, size = 73, normalized size = 0.40 \begin {gather*} \frac {2 (a+b x)^{3/2} (c+d x)^{5/4} \, _2F_1\left (-\frac {5}{4},\frac {3}{2};\frac {5}{2};\frac {d (a+b x)}{-b c+a d}\right )}{3 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 = 0.00, size = 0, normalized size = 0.00 \[\int \sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{4}}\, dx\]
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 [A]
time = 8.33, size = 218, normalized size = 1.20 \begin {gather*} - \frac {2 a d \left (a + b x\right )^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {a d e^{i \pi }}{b \operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}} + \frac {d x e^{i \pi }}{\operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}}} \right )} \sqrt [4]{\operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}}}{3 b^{2}} + \frac {2 c \left (a + b x\right )^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {a d e^{i \pi }}{b \operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}} + \frac {d x e^{i \pi }}{\operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}}} \right )} \sqrt [4]{\operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}}}{3 b} + \frac {2 d \left (a + b x\right )^{\frac {5}{2}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a d e^{i \pi }}{b \operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}} + \frac {d x e^{i \pi }}{\operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}}} \right )} \sqrt [4]{\operatorname {polar\_lift}{\left (- \frac {a d}{b} + c \right )}}}{5 b^{2}} \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 = -0.01 \begin {gather*} \int \sqrt {a+b\,x}\,{\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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