Optimal. Leaf size=163 \[ -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (3 b c-2 a d)}{c d^2}-\frac {2 (a+b x)^{3/2} (b c-a d)}{c d \sqrt {c+d x}} \]
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Rubi [A] time = 0.14, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {98, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (3 b c-2 a d)}{c d^2}-\frac {2 (a+b x)^{3/2} (b c-a d)}{c d \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 98
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x (c+d x)^{3/2}} \, dx &=-\frac {2 (b c-a d) (a+b x)^{3/2}}{c d \sqrt {c+d x}}+\frac {2 \int \frac {\sqrt {a+b x} \left (\frac {a^2 d}{2}+\frac {1}{2} b (3 b c-2 a d) x\right )}{x \sqrt {c+d x}} \, dx}{c d}\\ &=-\frac {2 (b c-a d) (a+b x)^{3/2}}{c d \sqrt {c+d x}}+\frac {b (3 b c-2 a d) \sqrt {a+b x} \sqrt {c+d x}}{c d^2}+\frac {2 \int \frac {\frac {a^3 d^2}{2}-\frac {1}{4} b^2 c (3 b c-5 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c d^2}\\ &=-\frac {2 (b c-a d) (a+b x)^{3/2}}{c d \sqrt {c+d x}}+\frac {b (3 b c-2 a d) \sqrt {a+b x} \sqrt {c+d x}}{c d^2}+\frac {a^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c}-\frac {\left (b^2 (3 b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2}\\ &=-\frac {2 (b c-a d) (a+b x)^{3/2}}{c d \sqrt {c+d x}}+\frac {b (3 b c-2 a d) \sqrt {a+b x} \sqrt {c+d x}}{c d^2}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c}-\frac {(b (3 b c-5 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^2}\\ &=-\frac {2 (b c-a d) (a+b x)^{3/2}}{c d \sqrt {c+d x}}+\frac {b (3 b c-2 a d) \sqrt {a+b x} \sqrt {c+d x}}{c d^2}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {(b (3 b c-5 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^2}\\ &=-\frac {2 (b c-a d) (a+b x)^{3/2}}{c d \sqrt {c+d x}}+\frac {b (3 b c-2 a d) \sqrt {a+b x} \sqrt {c+d x}}{c d^2}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.66, size = 226, normalized size = 1.39 \begin {gather*} \frac {2 \left (5 a \left (-\frac {a^{3/2} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {b \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2}}+\frac {\sqrt {a+b x} (a d-b c)}{c d}\right )+\frac {(a+b x)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{a d-b c}\right )}{c+d x}\right )}{5 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 229, normalized size = 1.40 \begin {gather*} -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {\sqrt {a+b x} \left (\frac {2 a^2 d^3 (a+b x)}{c+d x}-2 a^2 b d^2+\frac {2 b^2 c^2 d (a+b x)}{c+d x}+5 a b^2 c d-\frac {4 a b c d^2 (a+b x)}{c+d x}-3 b^3 c^2\right )}{c d^2 \sqrt {c+d x} \left (\frac {d (a+b x)}{c+d x}-b\right )}+\frac {\left (5 a b^{3/2} d-3 b^{5/2} c\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.11, size = 1181, normalized size = 7.25
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.68, size = 256, normalized size = 1.57 \begin {gather*} -\frac {2 \, \sqrt {b d} a^{3} b \arctan \left (-\frac {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}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c {\left | b \right |}} + \frac {\sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} b^{3}}{d {\left | b \right |}} + \frac {3 \, b^{6} c^{2} d - 5 \, a b^{5} c d^{2} + 2 \, a^{2} b^{4} d^{3}}{b^{2} c d^{3} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, \sqrt {b d} b^{3} c - 5 \, \sqrt {b d} a b^{2} 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 )}^{2}\right )}{2 \, d^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 492, normalized size = 3.02 \begin {gather*} \frac {\sqrt {b x +a}\, \left (-2 \sqrt {b d}\, a^{3} d^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+5 \sqrt {a c}\, a \,b^{2} 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 )-3 \sqrt {a c}\, b^{3} 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 )-2 \sqrt {b d}\, a^{3} c \,d^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+5 \sqrt {a c}\, a \,b^{2} 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 )-3 \sqrt {a c}\, b^{3} 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 )+2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d x +4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}-8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d +6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {d x +c}\, c \,d^{2}} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x\,{\left (c+d\,x\right )}^{3/2}} \,d x \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}}}{x \left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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