Optimal. Leaf size=101 \[ \frac {16 d^2 \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^3}+\frac {8 d}{3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac {16 d^2 \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^3}+\frac {8 d}{3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}-\frac {(4 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{3 (b c-a d)}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}+\frac {8 d}{3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {\left (8 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}+\frac {8 d}{3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 0.74 \[ \frac {2 \left (3 a^2 d^2+6 a b d (c+2 d x)+b^2 \left (-c^2+4 c d x+8 d^2 x^2\right )\right )}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 273, normalized size = 2.70 \[ \frac {2 \, {\left (8 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} + 4 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.54, size = 368, normalized size = 3.64 \[ \frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{{\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {4 \, {\left (5 \, \sqrt {b d} b^{6} c^{2} d - 10 \, \sqrt {b d} a b^{5} c d^{2} + 5 \, \sqrt {b d} a^{2} b^{4} d^{3} - 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c d + 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} d^{2} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} d\right )}}{3 \, {\left (b^{2} c^{2} {\left | b \right |} - 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 1.04 \[ -\frac {2 \left (8 b^{2} x^{2} d^{2}+12 a b \,d^{2} x +4 b^{2} c d x +3 a^{2} d^{2}+6 a b c d -b^{2} c^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 141, normalized size = 1.40 \[ -\frac {\sqrt {c+d\,x}\,\left (\frac {8\,x\,\left (3\,a\,d+b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,b\,d\,x^2}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {6\,a^2\,d^2+12\,a\,b\,c\,d-2\,b^2\,c^2}{3\,b\,d\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a\,c\,\sqrt {a+b\,x}}{b\,d}+\frac {x\,\left (a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}} \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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