Optimal. Leaf size=174 \[ -\frac {2 c (a+b x)^{5/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 (5 b c-3 a d) (a+b x)^{3/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {b (5 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {\sqrt {b} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}} \]
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Rubi [A]
time = 0.06, 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 = {79, 49, 52, 65,
223, 212} \begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
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) \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 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) \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 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 [A]
time = 0.23, size = 111, normalized size = 0.64 \begin {gather*} \frac {\sqrt {a+b x} \left (-2 a d (2 c+3 d x)+b \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 d^3 (c+d x)^{3/2}}-\frac {\sqrt {b} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs.
\(2(146)=292\).
time = 0.07, size = 459, normalized size = 2.64
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (9 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,d^{3} x^{2}-15 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c \,d^{2} x^{2}+18 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x -30 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d x +6 b \,d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3}-12 a \,d^{2} x \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+40 b c d x \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-8 a c d \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+30 b \,c^{2} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{6 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, d^{3} \left (d x +c \right )^{\frac {3}{2}}}\) | \(459\) |
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 [A]
time = 1.65, size = 431, normalized size = 2.48 \begin {gather*} \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 ] \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 {x \left (a + b x\right )^{\frac {3}{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 [A]
time = 1.19, size = 286, normalized size = 1.64 \begin {gather*} \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}} \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 {x\,{\left (a+b\,x\right )}^{3/2}}{{\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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