Optimal. Leaf size=750 \[ \frac {2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt {b c-a d} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {8 d (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)^2}+\frac {8 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3 (a d+b c+2 b d x) \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}-\frac {4 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt {b c-a d} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}} \]
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Rubi [A] time = 0.74, antiderivative size = 750, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {51, 62, 623, 305, 220, 1196} \[ -\frac {8 d (a+b x)^{3/4}}{\sqrt [4]{c+d x} (b c-a d)^2}+\frac {8 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3 (a d+b c+2 b d x) \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )}-\frac {4}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)}+\frac {2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt {b c-a d} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {4 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a+b x} \sqrt [4]{c+d x} \sqrt {b c-a d} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 62
Rule 220
Rule 305
Rule 623
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/4} (c+d x)^{5/4}} \, dx &=-\frac {4}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {(2 d) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{5/4}} \, dx}{b c-a d}\\ &=-\frac {4}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {8 d (a+b x)^{3/4}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {(4 b d) \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx}{(b c-a d)^2}\\ &=-\frac {4}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {8 d (a+b x)^{3/4}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (4 b d \sqrt [4]{(a+b x) (c+d x)}\right ) \int \frac {1}{\sqrt [4]{a c+(b c+a d) x+b d x^2}} \, dx}{(b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}\\ &=-\frac {4}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {8 d (a+b x)^{3/4}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (16 b d \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\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 )}{(b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac {4}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {8 d (a+b x)^{3/4}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (8 \sqrt {b} \sqrt {d} \sqrt [4]{(a+b x) (c+d x)} \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 )}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}-\frac {\left (8 \sqrt {b} \sqrt {d} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {2 \sqrt {b} \sqrt {d} x^2}{b c-a d}}{\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 )}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}\\ &=-\frac {4}{(b c-a d) \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {8 d (a+b x)^{3/4}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {8 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{(b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )}-\frac {4 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)} \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}} E\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 )}{\sqrt {b c-a d} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)} \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 )}{\sqrt {b c-a d} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (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 = 0.05, size = 71, normalized size = 0.09 \[ -\frac {4 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (-\frac {1}{4},\frac {5}{4};\frac {3}{4};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt [4]{a+b x} (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{b^{2} d^{2} x^{4} + a^{2} c^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} x^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} {\left (d x + c\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {5}{4}} \left (d x +c \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} {\left (d x + c\right )}^{\frac {5}{4}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,x\right )}^{5/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{4}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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