Optimal. Leaf size=101 \[ -\frac {2 \sqrt {c+d x}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {8 d \sqrt {c+d x}}{15 (b c-a d)^2 (a+b x)^{3/2}}-\frac {16 d^2 \sqrt {c+d x}}{15 (b c-a d)^3 \sqrt {a+b x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} -\frac {16 d^2 \sqrt {c+d x}}{15 \sqrt {a+b x} (b c-a d)^3}+\frac {8 d \sqrt {c+d x}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{5 (a+b x)^{5/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx &=-\frac {2 \sqrt {c+d x}}{5 (b c-a d) (a+b x)^{5/2}}-\frac {(4 d) \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx}{5 (b c-a d)}\\ &=-\frac {2 \sqrt {c+d x}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {8 d \sqrt {c+d x}}{15 (b c-a d)^2 (a+b x)^{3/2}}+\frac {\left (8 d^2\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{15 (b c-a d)^2}\\ &=-\frac {2 \sqrt {c+d x}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {8 d \sqrt {c+d x}}{15 (b c-a d)^2 (a+b x)^{3/2}}-\frac {16 d^2 \sqrt {c+d x}}{15 (b c-a d)^3 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 75, normalized size = 0.74 \begin {gather*} -\frac {2 \sqrt {c+d x} \left (15 a^2 d^2-10 a b d (c-2 d x)+b^2 \left (3 c^2-4 c d x+8 d^2 x^2\right )\right )}{15 (b c-a d)^3 (a+b x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 95, normalized size = 0.94
method | result | size |
default | \(-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\) | \(95\) |
gosper | \(\frac {2 \sqrt {d x +c}\, \left (8 b^{2} x^{2} d^{2}+20 a b \,d^{2} x -4 b^{2} c d x +15 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right )}{15 \left (b x +a \right )^{\frac {5}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 251 vs.
\(2 (83) = 166\).
time = 0.46, size = 251, normalized size = 2.49 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 4 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\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 {1}{\left (a + b x\right )^{\frac {7}{2}} \sqrt {c + d 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. 227 vs.
\(2 (83) = 166\).
time = 0.03, size = 254, normalized size = 2.51 \begin {gather*} -\frac {128 b \sqrt {b d} \left (b d\right )^{2} \left (-10 \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{4}-5 \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2} a b d+5 \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2} b^{2} c-a^{2} b^{2} d^{2}+2 a b^{3} d c-b^{4} c^{2}\right )}{60 \left |b\right | \left (\left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2}+a b d-b^{2} c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.01, size = 133, normalized size = 1.32 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {16\,d^2\,x^2}{15\,{\left (a\,d-b\,c\right )}^3}+\frac {30\,a^2\,d^2-20\,a\,b\,c\,d+6\,b^2\,c^2}{15\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d\,x\,\left (5\,a\,d-b\,c\right )}{15\,b\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a^2\,\sqrt {a+b\,x}}{b^2}+\frac {2\,a\,x\,\sqrt {a+b\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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