Optimal. Leaf size=191 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {101, 156, 12,
95, 214} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d) (a d+3 b c)}{24 a^3 c^2 x}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{12 a^2 c x^2}+\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {\int \frac {\frac {1}{2} (-5 b c+a d)-2 b d x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {\int \frac {-\frac {1}{4} (5 b c-3 a d) (3 b c+a d)-\frac {1}{2} b d (5 b c-a d) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^2 c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {\int -\frac {3 (b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^3 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^3 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^3 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 10.10, size = 160, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 b^2 c^2 x^2+2 a b c x (5 c+2 d x)+a^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )}{24 a^3 c^2 x^3}-\frac {\left (-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}} \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. \(407\) vs.
\(2(159)=318\).
time = 0.08, size = 408, normalized size = 2.14
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-8 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}+30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}+4 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x +16 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{3} c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}}\) | \(408\) |
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 = 1.12, size = 440, normalized size = 2.30 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \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 + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{3} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{3} x^{3}}\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 {\sqrt {c + d x}}{x^{4} \sqrt {a + b x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2140 vs.
\(2 (159) = 318\).
time = 6.53, size = 2140, normalized size = 11.20 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 91.00, size = 1574, normalized size = 8.24 \begin {gather*} \frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (\frac {5\,a^4\,b^2\,d^4}{64}+\frac {a^3\,b^3\,c\,d^3}{2}+\frac {7\,a^2\,b^4\,c^2\,d^2}{16}-\frac {23\,a\,b^5\,c^3\,d}{16}-\frac {13\,b^6\,c^4}{64}\right )}{a^4\,c^3\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {b^6}{192\,a^2\,c\,d^3}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (-\frac {a^5\,b\,d^5}{64}+\frac {7\,a^4\,b^2\,c\,d^4}{16}+\frac {37\,a^3\,b^3\,c^2\,d^3}{32}-\frac {47\,a^2\,b^4\,c^3\,d^2}{32}-\frac {3\,a\,b^5\,c^4\,d}{2}+\frac {17\,b^6\,c^5}{64}\right )}{a^{9/2}\,c^{7/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,b^4\,d^2}{64}+\frac {3\,a\,b^5\,c\,d}{32}-\frac {5\,b^6\,c^2}{64}\right )}{a^3\,c^2\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {3\,a^5\,d^5}{64}-\frac {13\,a^4\,b\,c\,d^4}{64}-\frac {49\,a^3\,b^2\,c^2\,d^3}{64}+\frac {69\,a^2\,b^3\,c^3\,d^2}{64}+\frac {45\,a\,b^4\,c^4\,d}{64}-\frac {27\,b^5\,c^5}{64}\right )}{a^{9/2}\,c^{7/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {\left (\frac {b^6\,c}{64}-\frac {a\,b^5\,d}{64}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^{5/2}\,c^{3/2}\,d^3\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (-\frac {17\,a^3\,b^3\,d^3}{192}-\frac {9\,a^2\,b^4\,c\,d^2}{32}+\frac {3\,a\,b^5\,c^2\,d}{32}+\frac {73\,b^6\,c^3}{192}\right )}{a^{7/2}\,c^{5/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (-\frac {5\,a^6\,d^6}{192}+\frac {a^5\,b\,c\,d^5}{16}+a^4\,b^2\,c^2\,d^4+\frac {5\,a^3\,b^3\,c^3\,d^3}{32}-\frac {5\,a^2\,b^4\,c^4\,d^2}{2}+\frac {a\,b^5\,c^5\,d}{8}+\frac {37\,b^6\,c^6}{192}\right )}{a^5\,c^4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (-\frac {a^4\,d^4}{64}+\frac {3\,a^3\,b\,c\,d^3}{32}+\frac {13\,a^2\,b^2\,c^2\,d^2}{64}-\frac {19\,a\,b^3\,c^3\,d}{32}+\frac {15\,b^4\,c^4}{64}\right )}{a^4\,c^3\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}+\frac {b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (3\,a^2\,d^2+9\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (a^3\,d^3+9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{a^{3/2}\,c^{3/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}-\frac {\left (3\,a\,d+3\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {\left (3\,c\,b^3+3\,a\,d\,b^2\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{\sqrt {a}\,\sqrt {c}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (3\,a^2\,b\,d^2+9\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{a\,c\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}}+\frac {\left (\frac {d^3}{32\,a^{3/2}\,c^{3/2}}-\frac {d^2\,\left (a\,d+b\,c\right )}{32\,a^{5/2}\,c^{3/2}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {d\,\left (3\,a^2\,d^2+8\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,a^3\,c^2}-\frac {4\,a\,d^7-3\,b\,c\,d^6}{64\,a^2\,c^2\,d^4}+\frac {4\,\left (\frac {d^3}{16\,a^{3/2}\,c^{3/2}}-\frac {d^2\,\left (a\,d+b\,c\right )}{16\,a^{5/2}\,c^{3/2}}\right )\,\left (a\,d+b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d}\right )}{\sqrt {c+d\,x}-\sqrt {c}}+\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (a^{7/2}\,\sqrt {c}\,d^3-5\,\sqrt {a}\,b^3\,c^{7/2}+3\,a^{3/2}\,b^2\,c^{5/2}\,d+a^{5/2}\,b\,c^{3/2}\,d^2\right )}{16\,a^4\,c^3}-\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (a^{7/2}\,\sqrt {c}\,d^3-5\,\sqrt {a}\,b^3\,c^{7/2}+3\,a^{3/2}\,b^2\,c^{5/2}\,d+a^{5/2}\,b\,c^{3/2}\,d^2\right )}{16\,a^4\,c^3}-\frac {d^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{192\,a^2\,c\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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