Optimal. Leaf size=222 \[ \frac {5 \sqrt {b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}-\frac {5 b \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{4 d^4}+\frac {5 b (a+b x)^{3/2} \sqrt {c+d x} (7 b c-3 a d)}{6 d^3 (b c-a d)}-\frac {2 (a+b x)^{5/2} (7 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 c (a+b x)^{7/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.12, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac {2 (a+b x)^{5/2} (7 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}+\frac {5 b (a+b x)^{3/2} \sqrt {c+d x} (7 b c-3 a d)}{6 d^3 (b c-a d)}-\frac {5 b \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{4 d^4}+\frac {5 \sqrt {b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}-\frac {2 c (a+b x)^{7/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {(7 b c-3 a d) \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx}{3 d (b c-a d)}\\ &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (7 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b (7 b c-3 a d)) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)}\\ &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (7 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {5 b (7 b c-3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)}-\frac {(5 b (7 b c-3 a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d^3}\\ &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (7 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 b (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {5 b (7 b c-3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)}+\frac {(5 b (7 b c-3 a d) (b c-a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^4}\\ &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (7 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 b (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {5 b (7 b c-3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)}+\frac {(5 (7 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 d^4}\\ &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (7 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 b (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {5 b (7 b c-3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)}+\frac {(5 (7 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 d^4}\\ &=-\frac {2 c (a+b x)^{7/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (7 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 b (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {5 b (7 b c-3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)}+\frac {5 \sqrt {b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 110, normalized size = 0.50 \[ \frac {2 (a+b x)^{7/2} \left ((c+d x) (7 b c-3 a d) \sqrt {\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {d (a+b x)}{a d-b c}\right )+7 c (a d-b c)\right )}{21 d (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.65, size = 619, normalized size = 2.79 \[ \left [\frac {15 \, {\left (7 \, b^{2} c^{4} - 10 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (7 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (7 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (6 \, b^{2} d^{3} x^{3} - 105 \, b^{2} c^{3} + 115 \, a b c^{2} d - 16 \, a^{2} c d^{2} - 3 \, {\left (7 \, b^{2} c d^{2} - 9 \, a b d^{3}\right )} x^{2} - 2 \, {\left (70 \, b^{2} c^{2} d - 79 \, a b c d^{2} + 12 \, a^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, -\frac {15 \, {\left (7 \, b^{2} c^{4} - 10 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (7 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (7 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - 2 \, {\left (6 \, b^{2} d^{3} x^{3} - 105 \, b^{2} c^{3} + 115 \, a b c^{2} d - 16 \, a^{2} c d^{2} - 3 \, {\left (7 \, b^{2} c d^{2} - 9 \, a b d^{3}\right )} x^{2} - 2 \, {\left (70 \, b^{2} c^{2} d - 79 \, a b c d^{2} + 12 \, a^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.61, size = 404, normalized size = 1.82 \[ \frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{5} c d^{6} {\left | b \right |} - a b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c d^{7} - a b^{3} d^{8}} - \frac {7 \, b^{6} c^{2} d^{5} {\left | b \right |} - 10 \, a b^{5} c d^{6} {\left | b \right |} + 3 \, a^{2} b^{4} d^{7} {\left | b \right |}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} - \frac {20 \, {\left (7 \, b^{7} c^{3} d^{4} {\left | b \right |} - 17 \, a b^{6} c^{2} d^{5} {\left | b \right |} + 13 \, a^{2} b^{5} c d^{6} {\left | b \right |} - 3 \, a^{3} b^{4} d^{7} {\left | b \right |}\right )}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, b^{8} c^{4} d^{3} {\left | b \right |} - 24 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 30 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 16 \, a^{3} b^{5} c d^{6} {\left | b \right |} + 3 \, a^{4} b^{4} d^{7} {\left | b \right |}\right )}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, b^{2} c^{2} {\left | b \right |} - 10 \, a b c d {\left | b \right |} + 3 \, a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 750, normalized size = 3.38 \[ \frac {\sqrt {b x +a}\, \left (45 a^{2} b \,d^{4} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-150 a \,b^{2} c \,d^{3} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+105 b^{3} c^{2} d^{2} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+90 a^{2} b c \,d^{3} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-300 a \,b^{2} c^{2} d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+210 b^{3} c^{3} d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+45 a^{2} b \,c^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-150 a \,b^{2} c^{3} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+105 b^{3} c^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d^{3} x^{3}+54 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,d^{3} x^{2}-42 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c \,d^{2} x^{2}-48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{3} x +316 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b c \,d^{2} x -280 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c^{2} d x -32 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{2}+230 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,c^{2} d -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c^{3}\right )}{24 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \left (d x +c \right )^{\frac {3}{2}} d^{4}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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