Optimal. Leaf size=111 \[ -\frac {2 \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a+b x} (b c-a d)^{3/4}}-\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)} \]
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Rubi [A] time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {51, 63, 224, 221} \[ -\frac {2 \sqrt [4]{c+d x}}{\sqrt {a+b x} (b c-a d)}-\frac {2 \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {a+b x} (b c-a d)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/4}} \, dx &=-\frac {2 \sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+b x}}-\frac {d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{2 (b c-a d)}\\ &=-\frac {2 \sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+b x}}-\frac {2 \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 )}{b c-a d}\\ &=-\frac {2 \sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+b x}}-\frac {\left (2 \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 )}{(b c-a d) \sqrt {a+b x}}\\ &=-\frac {2 \sqrt [4]{c+d x}}{(b c-a d) \sqrt {a+b x}}-\frac {2 \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 )}{\sqrt [4]{b} (b c-a d)^{3/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 71, normalized size = 0.64 \[ -\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt {a+b x} (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{4}}}{b^{2} d x^{3} + a^{2} c + {\left (b^{2} c + 2 \, a b d\right )} x^{2} + {\left (2 \, a b c + a^{2} d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/4}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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