Optimal. Leaf size=219 \[ \frac {\left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d) (a d+b c)}{8 b^2}-2 \sqrt {a} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+5 b c)}{12 b} \]
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Rubi [A] time = 0.22, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} \frac {\left (-5 a^2 b c d^2+a^3 d^3+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d) (a d+b c)}{8 b^2}-2 \sqrt {a} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+5 b c)}{12 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 101
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x} \, dx &=\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}-\frac {1}{3} \int \frac {(c+d x)^{3/2} \left (-3 a c+\frac {1}{2} (-5 b c-a d) x\right )}{x \sqrt {a+b x}} \, dx\\ &=\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b}+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}-\frac {\int \frac {\sqrt {c+d x} \left (-6 a b c^2-\frac {3}{4} (5 b c-a d) (b c+a d) x\right )}{x \sqrt {a+b x}} \, dx}{6 b}\\ &=\frac {(5 b c-a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2}+\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b}+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}-\frac {\int \frac {-6 a b^2 c^3-\frac {3}{8} \left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b^2}\\ &=\frac {(5 b c-a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2}+\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b}+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}+\left (a c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^2}\\ &=\frac {(5 b c-a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2}+\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b}+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}+\left (2 a c^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 (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\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 )}{8 b^3}\\ &=\frac {(5 b c-a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2}+\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b}+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}-2 \sqrt {a} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\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 )}{8 b^3}\\ &=\frac {(5 b c-a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2}+\frac {(5 b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b}+\frac {1}{3} \sqrt {a+b x} (c+d x)^{5/2}-2 \sqrt {a} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 274, normalized size = 1.25 \begin {gather*} \frac {\sqrt {c+d x} \left (-\frac {b \sqrt {d} \left (\sqrt {a+b x} (c+d x) \left (-3 a^2 d^2+2 a b d (7 c+d x)+b^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )-48 \sqrt {a} b^2 c^{5/2} \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}-3 \sqrt {b c-a d} \left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{24 b^2 \sqrt {d} (a d-b c) \sqrt {\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.63, size = 396, normalized size = 1.81 \begin {gather*} \frac {\left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} \sqrt {d}}+\frac {\sqrt {a+b x} \left (-3 a^3 b^2 d^3+\frac {3 a^3 d^5 (a+b x)^2}{(c+d x)^2}-\frac {8 a^3 b d^4 (a+b x)}{c+d x}+15 a^2 b^3 c d^2+\frac {24 a^2 b^2 c d^3 (a+b x)}{c+d x}-\frac {15 a^2 b c d^4 (a+b x)^2}{(c+d x)^2}-\frac {40 b^4 c^3 d (a+b x)}{c+d x}-45 a b^4 c^2 d+\frac {15 b^3 c^3 d^2 (a+b x)^2}{(c+d x)^2}+\frac {24 a b^3 c^2 d^2 (a+b x)}{c+d x}-\frac {3 a b^2 c^2 d^3 (a+b x)^2}{(c+d x)^2}+33 b^5 c^3\right )}{24 b^2 \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-2 \sqrt {a} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 19.51, size = 1197, normalized size = 5.47
result too large to display
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 = 583, normalized size = 2.66 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \sqrt {a c}\, a^{3} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-15 \sqrt {a c}\, a^{2} b c \,d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-48 \sqrt {b d}\, a \,b^{2} c^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+45 \sqrt {a c}\, a \,b^{2} c^{2} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+15 \sqrt {a c}\, b^{3} c^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+16 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} d^{2} x^{2}+4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a b \,d^{2} x +52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} d^{2}+28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a b c d +66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{2} c^{2}\right )}{48 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{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 = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{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 {\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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