Optimal. Leaf size=174 \[ -\frac {\sqrt {b} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d)}{d^3 (b c-a d)}-\frac {2 (a+b x)^{3/2} (5 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 c (a+b x)^{5/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.09, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {2 (a+b x)^{3/2} (5 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d)}{d^3 (b c-a d)}-\frac {\sqrt {b} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}}-\frac {2 c (a+b x)^{5/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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)^{3/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {(5 b c-3 a d) \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx}{3 d (b c-a d)}\\ &=-\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (5 b c-3 a d) (a+b x)^{3/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {(b (5 b c-3 a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)}\\ &=-\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (5 b c-3 a d) (a+b x)^{3/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {b (5 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(b (5 b c-3 a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^3}\\ &=-\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (5 b c-3 a d) (a+b x)^{3/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {b (5 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-3 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 )}{d^3}\\ &=-\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (5 b c-3 a d) (a+b x)^{3/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {b (5 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-3 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 )}{d^3}\\ &=-\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (5 b c-3 a d) (a+b x)^{3/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {b (5 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {\sqrt {b} (5 b c-3 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 [C] time = 0.14, size = 110, normalized size = 0.63 \[ \frac {2 (a+b x)^{5/2} \left ((c+d x) (5 b c-3 a d) \sqrt {\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{a d-b c}\right )+5 c (a d-b c)\right )}{15 d (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 431, normalized size = 2.48 \[ \left [-\frac {3 \, {\left (5 \, b c^{3} - 3 \, a c^{2} d + {\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} x^{2} + 2 \, {\left (5 \, b c^{2} d - 3 \, a c d^{2}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b d^{2} x^{2} + 15 \, b c^{2} - 4 \, a c d + 2 \, {\left (10 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, \frac {3 \, {\left (5 \, b c^{3} - 3 \, a c^{2} d + {\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} x^{2} + 2 \, {\left (5 \, b c^{2} d - 3 \, a c d^{2}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b d^{2} x^{2} + 15 \, b c^{2} - 4 \, a c d + 2 \, {\left (10 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.89, size = 286, normalized size = 1.64 \[ \frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{5} c d^{4} {\left | b \right |} - a b^{4} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c d^{5} - a b^{3} d^{6}} + \frac {4 \, {\left (5 \, b^{6} c^{2} d^{3} {\left | b \right |} - 8 \, a b^{5} c d^{4} {\left | b \right |} + 3 \, a^{2} b^{4} d^{5} {\left | b \right |}\right )}}{b^{4} c d^{5} - a b^{3} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{7} c^{3} d^{2} {\left | b \right |} - 13 \, a b^{6} c^{2} d^{3} {\left | b \right |} + 11 \, a^{2} b^{5} c d^{4} {\left | b \right |} - 3 \, a^{3} b^{4} d^{5} {\left | b \right |}\right )}}{b^{4} c d^{5} - a b^{3} d^{6}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b c {\left | b \right |} - 3 \, a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 459, normalized size = 2.64 \[ \frac {\sqrt {b x +a}\, \left (9 a b \,d^{3} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-15 b^{2} c \,d^{2} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+18 a b c \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-30 b^{2} c^{2} d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+9 a b \,c^{2} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-15 b^{2} c^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,d^{2} x^{2}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,d^{2} x +40 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b c d x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a c d +30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2}\right )}{6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} d^{3}} \]
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 = -0.01 \[ \int \frac {x\,{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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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