Optimal. Leaf size=137 \[ -\frac {16 b^{3/4} (b c-a d)^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d^3 \sqrt {a+b x}}+\frac {8 b \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}} \]
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Rubi [A] time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {47, 50, 63, 224, 221} \[ -\frac {16 b^{3/4} (b c-a d)^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^3 \sqrt {a+b x}}+\frac {8 b \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 221
Rule 224
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{(c+d x)^{7/4}} \, dx &=-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac {(2 b) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx}{d}\\ &=-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac {8 b \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac {(4 b (b c-a d)) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{3 d^2}\\ &=-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac {8 b \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac {(16 b (b c-a d)) \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 )}{3 d^3}\\ &=-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac {8 b \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac {\left (16 b (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 )}{3 d^3 \sqrt {a+b x}}\\ &=-\frac {4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac {8 b \sqrt {a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac {16 b^{3/4} (b c-a d)^{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 )}{3 d^3 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 73, normalized size = 0.53 \[ \frac {2 (a+b x)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (\frac {7}{4},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{a d-b c}\right )}{5 b (c+d x)^{7/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{4}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (d x +c \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {7}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{7/4}} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\left (c + d x\right )^{\frac {7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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