Optimal. Leaf size=184 \[ -\frac {6 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{7/4} \sqrt {a+b x}}+\frac {6 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt {a+b x}}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}} \]
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Rubi [A] time = 0.21, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {47, 63, 307, 224, 221, 1200, 1199, 424} \[ -\frac {6 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt {a+b x}}+\frac {6 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt {a+b x}}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 221
Rule 224
Rule 307
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx &=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}+\frac {(3 d) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{2 b}\\ &=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}+\frac {6 \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 )}{b}\\ &=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}-\frac {\left (6 \sqrt {b c-a d}\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 )}{b^{3/2}}+\frac {\left (6 \sqrt {b c-a d}\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 )}{b^{3/2}}\\ &=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}-\frac {\left (6 \sqrt {b c-a d} \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^{3/2} \sqrt {a+b x}}+\frac {\left (6 \sqrt {b c-a d} \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 )}{b^{3/2} \sqrt {a+b x}}\\ &=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}-\frac {6 (b c-a d)^{3/4} \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 )}{b^{7/4} \sqrt {a+b x}}+\frac {\left (6 \sqrt {b c-a d} \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 )}{b^{3/2} \sqrt {a+b x}}\\ &=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}+\frac {6 (b c-a d)^{3/4} \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 )}{b^{7/4} \sqrt {a+b x}}-\frac {6 (b c-a d)^{3/4} \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 )}{b^{7/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 71, normalized size = 0.39 \[ -\frac {2 (c+d x)^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^{3/4}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (c + d x\right )^{\frac {3}{4}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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