Optimal. Leaf size=270 \[ \frac {3 d^2 \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{10 b^{7/4} \sqrt {a+b x} (b c-a d)^{5/4}}-\frac {3 d^2 \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt {a+b x} (b c-a d)^{5/4}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b \sqrt {a+b x} (b c-a d)^2}-\frac {d (c+d x)^{3/4}}{5 b (a+b x)^{3/2} (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}} \]
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Rubi [A] time = 0.28, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {47, 51, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac {3 d^2 \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt {a+b x} (b c-a d)^{5/4}}-\frac {3 d^2 \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{10 b^{7/4} \sqrt {a+b x} (b c-a d)^{5/4}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b \sqrt {a+b x} (b c-a d)^2}-\frac {d (c+d x)^{3/4}}{5 b (a+b x)^{3/2} (b c-a d)}-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
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)^{7/2}} \, dx &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}+\frac {(3 d) \int \frac {1}{(a+b x)^{5/2} \sqrt [4]{c+d x}} \, dx}{10 b}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}-\frac {\left (3 d^2\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt [4]{c+d x}} \, dx}{20 b (b c-a d)}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b (b c-a d)^2 \sqrt {a+b x}}-\frac {\left (3 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{40 b (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b (b c-a d)^2 \sqrt {a+b x}}-\frac {\left (3 d^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 )}{10 b (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (3 d^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 )}{10 b^{3/2} (b c-a d)^{3/2}}-\frac {\left (3 d^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 )}{10 b^{3/2} (b c-a d)^{3/2}}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (3 d^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 )}{10 b^{3/2} (b c-a d)^{3/2} \sqrt {a+b x}}-\frac {\left (3 d^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 )}{10 b^{3/2} (b c-a d)^{3/2} \sqrt {a+b x}}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b (b c-a d)^2 \sqrt {a+b x}}+\frac {3 d^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 )}{10 b^{7/4} (b c-a d)^{5/4} \sqrt {a+b x}}-\frac {\left (3 d^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 )}{10 b^{3/2} (b c-a d)^{3/2} \sqrt {a+b x}}\\ &=-\frac {2 (c+d x)^{3/4}}{5 b (a+b x)^{5/2}}-\frac {d (c+d x)^{3/4}}{5 b (b c-a d) (a+b x)^{3/2}}+\frac {3 d^2 (c+d x)^{3/4}}{10 b (b c-a d)^2 \sqrt {a+b x}}-\frac {3 d^2 \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 )}{10 b^{7/4} (b c-a d)^{5/4} \sqrt {a+b x}}+\frac {3 d^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 )}{10 b^{7/4} (b c-a d)^{5/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.27 \[ -\frac {2 (c+d x)^{3/4} \, _2F_1\left (-\frac {5}{2},-\frac {3}{4};-\frac {3}{2};\frac {d (a+b x)}{a d-b c}\right )}{5 b (a+b x)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (d x +c \right )^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^{3/4}}{{\left (a+b\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (c + d x\right )^{\frac {3}{4}}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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