Optimal. Leaf size=186 \[ \frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d}+\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {52, 65, 223,
212} \begin {gather*} -\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{3/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 b^3 d}+\frac {5 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{32 b^3}+\frac {5 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \sqrt {a+b x} (c+d x)^{5/2} \, dx &=\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}+\frac {(5 (b c-a d)) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{8 b}\\ &=\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}+\frac {\left (5 (b c-a d)^2\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{16 b^2}\\ &=\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}+\frac {\left (5 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b^3}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d}+\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac {\left (5 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^3 d}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d}+\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac {\left (5 (b c-a d)^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^4 d}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d}+\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac {\left (5 (b c-a d)^4\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b^4 d}\\ &=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d}+\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 144, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 d^3 (a+b x)^3-55 b d^2 (a+b x)^2 (c+d x)+73 b^2 d (a+b x) (c+d x)^2+15 b^3 (c+d x)^3\right )}{192 b^3 d}-\frac {5 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 206, normalized size = 1.11
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {7}{2}}}{4 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {5}{2}} \sqrt {b x +a}}{3 b}-\frac {5 \left (a d -b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 b}\right )}{8 d}\) | \(206\) |
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 = 1.16, size = 540, normalized size = 2.90 \begin {gather*} \left [\frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {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 x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 73 \, a b^{3} c^{2} d^{2} - 55 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{4} d^{2}}, \frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 73 \, a b^{3} c^{2} d^{2} - 55 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{4} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 1083 vs.
\(2 (148) = 296\).
time = 1.22, size = 1083, normalized size = 5.82 \begin {gather*} -\frac {\frac {192 \, {\left (\frac {{\left (b^{2} c - a b 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 |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} a c^{2} {\left | b \right |}}{b^{2}} - \frac {16 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\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 |}\right )}{\sqrt {b d} b d^{2}}\right )} c d {\left | b \right |}}{b} - \frac {8 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\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 |}\right )}{\sqrt {b d} b d^{2}}\right )} a d^{2} {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\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 |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} d^{2} {\left | b \right |}}{b} - \frac {48 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\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 |}\right )}{\sqrt {b d} d}\right )} c^{2} {\left | b \right |}}{b^{2}} - \frac {96 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\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 |}\right )}{\sqrt {b d} d}\right )} a c d {\left | b \right |}}{b^{3}}}{192 \, b} \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 \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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