Optimal. Leaf size=218 \[ -2 a^{5/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-5 a d) (a d+b c)}{8 d^2}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (5 a d+b c)}{12 d} \]
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Rubi [A] time = 0.25, antiderivative size = 218, 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 (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{5/2}}-2 a^{5/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-5 a d) (a d+b c)}{8 d^2}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (5 a d+b c)}{12 d} \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 {(a+b x)^{5/2} \sqrt {c+d x}}{x} \, dx &=\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}-\frac {1}{3} \int \frac {(a+b x)^{3/2} \left (-3 a c+\frac {1}{2} (-b c-5 a d) x\right )}{x \sqrt {c+d x}} \, dx\\ &=\frac {(b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}-\frac {\int \frac {\sqrt {a+b x} \left (-6 a^2 c d+\frac {3}{4} (b c-5 a d) (b c+a d) x\right )}{x \sqrt {c+d x}} \, dx}{6 d}\\ &=-\frac {(b c-5 a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}+\frac {(b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}-\frac {\int \frac {-6 a^3 c d^2-\frac {3}{8} \left (16 a^2 b c d^2+(b c-5 a d) (b c-a d) (b c+a d)\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 d^2}\\ &=-\frac {(b c-5 a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}+\frac {(b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}+\left (a^3 c\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^2}\\ &=-\frac {(b c-5 a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}+\frac {(b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}+\left (2 a^3 c\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 (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 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 d^2}\\ &=-\frac {(b c-5 a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}+\frac {(b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}-2 a^{5/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 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 d^2}\\ &=-\frac {(b c-5 a d) (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 d^2}+\frac {(b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d}+\frac {1}{3} (a+b x)^{5/2} \sqrt {c+d x}-2 a^{5/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 242, normalized size = 1.11 \begin {gather*} \frac {\sqrt {d} \left (\sqrt {a+b x} (c+d x) \left (33 a^2 d^2+2 a b d (7 c+13 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )-48 a^{5/2} \sqrt {c} d^2 \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )+\frac {3 \sqrt {b c-a d} \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+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 )}{b}}{24 d^{5/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.57, size = 396, normalized size = 1.82 \begin {gather*} -2 a^{5/2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 \sqrt {b} d^{5/2}}+\frac {\sqrt {c+d x} \left (\frac {15 a^3 b^2 d^3 (c+d x)^2}{(a+b x)^2}-\frac {40 a^3 b d^4 (c+d x)}{a+b x}+33 a^3 d^5-\frac {3 a^2 b^3 c d^2 (c+d x)^2}{(a+b x)^2}+\frac {24 a^2 b^2 c d^3 (c+d x)}{a+b x}-45 a^2 b c d^4+\frac {3 b^5 c^3 (c+d x)^2}{(a+b x)^2}-\frac {8 b^4 c^3 d (c+d x)}{a+b x}-\frac {15 a b^4 c^2 d (c+d x)^2}{(a+b x)^2}+\frac {24 a b^3 c^2 d^2 (c+d x)}{a+b x}+15 a b^2 c^2 d^3-3 b^3 c^3 d^2\right )}{24 d^2 \sqrt {a+b x} \left (d-\frac {b (c+d x)}{a+b x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 19.89, size = 1197, normalized size = 5.49
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 = 2.67 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-48 \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 {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+15 \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 )+45 \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 )-15 \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 )+3 \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}+52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a b \,d^{2} x +4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{2} c d x +66 \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 -6 \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}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}}{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}} \sqrt {c + d x}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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