Optimal. Leaf size=128 \[ \frac {5 d^2 \sqrt {a+b x} \sqrt {c+d x}}{b^3}-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {5 d^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.05, 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 d^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}}+\frac {5 d^2 \sqrt {a+b x} \sqrt {c+d x}}{b^3}-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b 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 {(c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {(5 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx}{3 b}\\ &=-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {\left (5 d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{b^2}\\ &=\frac {5 d^2 \sqrt {a+b x} \sqrt {c+d x}}{b^3}-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {\left (5 d^2 (b c-a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^3}\\ &=\frac {5 d^2 \sqrt {a+b x} \sqrt {c+d x}}{b^3}-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {\left (5 d^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^4}\\ &=\frac {5 d^2 \sqrt {a+b x} \sqrt {c+d x}}{b^3}-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {\left (5 d^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^4}\\ &=\frac {5 d^2 \sqrt {a+b x} \sqrt {c+d x}}{b^3}-\frac {10 d (c+d x)^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac {5 d^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 121, normalized size = 0.95 \begin {gather*} \frac {\frac {\sqrt {c+d x} \left (15 a^2 d^2-10 a b d (c-2 d x)+b^2 \left (-2 c^2-14 c d x+3 d^2 x^2\right )\right )}{(a+b x)^{3/2}}+\frac {15 d (-b c+a d) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )}{\sqrt {\frac {b}{d}}}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {5}{2}}}{\left (b x +a \right )^{\frac {5}{2}}}\, dx\]
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 = 0.88, size = 475, normalized size = 3.71 \begin {gather*} \left [-\frac {15 \, {\left (a^{2} b c d - a^{3} d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 2 \, {\left (7 \, b^{2} c d - 10 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {15 \, {\left (a^{2} b c d - a^{3} d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 2 \, {\left (7 \, b^{2} c d - 10 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{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 (c + d x\right )^{\frac {5}{2}}}{\left (a + b 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. 650 vs.
\(2 (100) = 200\).
time = 1.21, size = 650, normalized size = 5.08 \begin {gather*} \frac {\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} d^{2} {\left | b \right |}}{b^{5}} - \frac {5 \, {\left (\sqrt {b d} b c d {\left | b \right |} - \sqrt {b d} a d^{2} {\left | b \right |}\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 )}^{2}\right )}{2 \, b^{5}} - \frac {4 \, {\left (7 \, \sqrt {b d} b^{6} c^{4} d {\left | b \right |} - 28 \, \sqrt {b d} a b^{5} c^{3} d^{2} {\left | b \right |} + 42 \, \sqrt {b d} a^{2} b^{4} c^{2} d^{3} {\left | b \right |} - 28 \, \sqrt {b d} a^{3} b^{3} c d^{4} {\left | b \right |} + 7 \, \sqrt {b d} a^{4} b^{2} d^{5} {\left | b \right |} - 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^{3} d {\left | b \right |} + 36 \, \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} c^{2} d^{2} {\left | b \right |} - 36 \, \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^{2} b^{2} c d^{3} {\left | b \right |} + 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^{3} b d^{4} {\left | b \right |} + 9 \, \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} c^{2} d {\left | b \right |} - 18 \, \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} a b c d^{2} {\left | b \right |} + 9 \, \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} a^{2} d^{3} {\left | b \right |}\right )}}{3 \, {\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} b^{4}} \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 (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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