Optimal. Leaf size=236 \[ \frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d \sqrt {a+b x} (b c-a d)^{5/4}}-\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt {a+b x} (b c-a d)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^2}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)} \]
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Rubi [A] time = 0.24, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {51, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt {a+b x} (b c-a d)^{5/4}}-\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt {a+b x} (b c-a d)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^2}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 221
Rule 224
Rule 307
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {(3 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx}{5 (b c-a d)}\\ &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{5 (b c-a d)^2}\\ &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d (b c-a d)^2}\\ &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (12 b^{3/2}\right ) \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 )}{5 d (b c-a d)^{3/2}}-\frac {\left (12 b^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d (b c-a d)^{3/2}}\\ &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (12 b^{3/2} \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 )}{5 d (b c-a d)^{3/2} \sqrt {a+b x}}-\frac {\left (12 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d (b c-a d)^{3/2} \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}+\frac {12 b^{5/4} \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 )}{5 d (b c-a d)^{5/4} \sqrt {a+b x}}-\frac {\left (12 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d (b c-a d)^{3/2} \sqrt {a+b x}}\\ &=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d (b c-a d)^{5/4} \sqrt {a+b x}}+\frac {12 b^{5/4} \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 )}{5 d (b c-a d)^{5/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 71, normalized size = 0.30 \[ \frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \, _2F_1\left (\frac {1}{2},\frac {9}{4};\frac {3}{2};\frac {d (a+b x)}{a d-b c}\right )}{b (c+d x)^{9/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}}{b d^{3} x^{4} + a c^{3} + {\left (3 \, b c d^{2} + a d^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} x^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {9}{4}}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{9/4}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {9}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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