Optimal. Leaf size=171 \[ -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{5/2}}-\frac {b \sqrt {a+b x} \sqrt {c+d x} (3 b c-7 a d)}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d} \]
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Rubi [A] time = 0.15, antiderivative size = 171, 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 = {102, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} \frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{5/2}}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {b \sqrt {a+b x} \sqrt {c+d x} (3 b c-7 a d)}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 102
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x \sqrt {c+d x}} \, dx &=\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\int \frac {\sqrt {a+b x} \left (2 a^2 d-\frac {1}{2} b (3 b c-7 a d) x\right )}{x \sqrt {c+d x}} \, dx}{2 d}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\int \frac {2 a^3 d^2+\frac {1}{4} b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d}+a^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\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 )+\frac {\left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \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^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \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^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {b (a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 197, normalized size = 1.15 \begin {gather*} \frac {1}{4} \left (-\frac {8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}+\frac {\left (-15 a^3 d^3+25 a^2 b c d^2-13 a b^2 c^2 d+3 b^3 c^3\right ) \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^{5/2} \sqrt {c+d x} \sqrt {b c-a d}}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (9 a d-3 b c+2 b d x)}{d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 248, normalized size = 1.45 \begin {gather*} -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {c}}+\frac {\left (15 a^2 \sqrt {b} d^2-10 a b^{3/2} c d+3 b^{5/2} c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 d^{5/2}}-\frac {b \sqrt {c+d x} \left (\frac {7 a^2 b d^2 (c+d x)}{a+b x}-9 a^2 d^3+\frac {3 b^3 c^2 (c+d x)}{a+b x}-\frac {10 a b^2 c d (c+d x)}{a+b x}+14 a b c d^2-5 b^2 c^2 d\right )}{4 d^2 \sqrt {a+b x} \left (d-\frac {b (c+d x)}{a+b x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 8.73, size = 987, normalized size = 5.77 \begin {gather*} \left [\frac {8 \, a^{2} d^{2} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\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 (2 \, b^{2} d x - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, d^{2}}, \frac {4 \, a^{2} d^{2} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\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 (2 \, b^{2} d x - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, d^{2}}, \frac {16 \, a^{2} d^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\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 (2 \, b^{2} d x - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, d^{2}}, \frac {8 \, a^{2} d^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\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 (2 \, b^{2} d x - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 342, normalized size = 2.00 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-8 \sqrt {b d}\, a^{3} 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 )+15 \sqrt {a c}\, a^{2} b \,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 )-10 \sqrt {a c}\, a \,b^{2} c 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^{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 )+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d x +18 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b d -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c \right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a 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\,\sqrt {c+d\,x}} \,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 \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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