Optimal. Leaf size=332 \[ \frac {5 (b c-a d)^{5/2} ((a+b x) (c+d x))^{3/4} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right ),\frac {1}{2}\right )}{6 \sqrt {2} \sqrt [4]{b} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {5 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}{3 d^2}+\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d} \]
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Rubi [A] time = 0.29, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 62, 623, 220} \[ -\frac {5 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}{3 d^2}+\frac {5 (b c-a d)^{5/2} ((a+b x) (c+d x))^{3/4} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{6 \sqrt {2} \sqrt [4]{b} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 62
Rule 220
Rule 623
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/4}}{(c+d x)^{3/4}} \, dx &=\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d}-\frac {(5 (b c-a d)) \int \frac {\sqrt [4]{a+b x}}{(c+d x)^{3/4}} \, dx}{6 d}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 d^2}+\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^{3/4} (c+d x)^{3/4}} \, dx}{12 d^2}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 d^2}+\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d}+\frac {\left (5 (b c-a d)^2 ((a+b x) (c+d x))^{3/4}\right ) \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^{3/4}} \, dx}{12 d^2 (a+b x)^{3/4} (c+d x)^{3/4}}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 d^2}+\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d}+\frac {\left (5 (b c-a d)^2 ((a+b x) (c+d x))^{3/4} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{3 d^2 (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x)}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 d^2}+\frac {2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{3 d}+\frac {5 (b c-a d)^{5/2} ((a+b x) (c+d x))^{3/4} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{6 \sqrt {2} \sqrt [4]{b} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.22 \[ \frac {4 (a+b x)^{9/4} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {9}{4};\frac {13}{4};\frac {d (a+b x)}{a d-b c}\right )}{9 b (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {5}{4}}}{\left (d x +c \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{5/4}}{{\left (c+d\,x\right )}^{3/4}} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {5}{4}}}{\left (c + d x\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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