Optimal. Leaf size=128 \[ -\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {49, 52, 65, 223,
212} \begin {gather*} -\frac {5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}+\frac {(5 b) \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {\left (5 b^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d^2}\\ &=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {\left (5 b^2 (b c-a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^3}\\ &=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {(5 b (b c-a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d^3}\\ &=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {(5 b (b c-a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d^3}\\ &=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 123, normalized size = 0.96 \begin {gather*} \frac {\frac {\sqrt {a+b x} \left (-2 a^2 d^2-2 a b d (5 c+7 d x)+b^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{(c+d x)^{3/2}}+15 b \sqrt {\frac {b}{d}} (b c-a d) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )}{3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (d x +c \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs.
\(2 (100) = 200\).
time = 0.84, size = 475, normalized size = 3.71 \begin {gather*} \left [-\frac {15 \, {\left (b^{2} c^{3} - a b c^{2} d + {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \, {\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, \frac {15 \, {\left (b^{2} c^{3} - a b c^{2} d + {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \, {\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 276 vs.
\(2 (100) = 200\).
time = 1.31, size = 276, normalized size = 2.16 \begin {gather*} \frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} {\left (b x + a\right )}}{b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}} + \frac {20 \, {\left (b^{7} c^{2} d^{3} - 2 \, a b^{6} c d^{4} + a^{2} b^{5} d^{5}\right )}}{b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}}\right )} + \frac {15 \, {\left (b^{8} c^{3} d^{2} - 3 \, a b^{7} c^{2} d^{3} + 3 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {5 \, {\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{3} {\left | b \right |}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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