Optimal. Leaf size=264 \[ -\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{7/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 157, 12,
95, 214} \begin {gather*} \frac {(5 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{7/2}}-\frac {d \sqrt {a+b x} \left (5 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {b (3 b c-a d)}{a^2 c \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {d \sqrt {a+b x} \left (-15 a^3 d^3+31 a^2 b c d^2-9 a b^2 c^2 d+9 b^3 c^3\right )}{3 a^2 c^3 \sqrt {c+d x} (b c-a d)^3}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {\int \frac {\frac {1}{2} (3 b c+5 a d)+3 b d x}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{a c}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \int \frac {\frac {1}{4} (b c-a d) (3 b c+5 a d)+b d (3 b c-a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{a^2 c (b c-a d)}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 \int \frac {-\frac {3}{8} (b c-a d)^2 (3 b c+5 a d)-\frac {1}{4} b d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {8 \int \frac {3 (b c-a d)^3 (3 b c+5 a d)}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(3 b c+5 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2 c^3}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(3 b c+5 a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^2 c^3}\\ &=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {1}{a c x \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (9 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \sqrt {a+b x}}{3 a^2 c^2 (b c-a d)^2 (c+d x)^{3/2}}-\frac {d \left (9 b^3 c^3-9 a b^2 c^2 d+31 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x}}{3 a^2 c^3 (b c-a d)^3 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 235, normalized size = 0.89 \begin {gather*} \frac {9 b^4 c^3 x (c+d x)^2+3 a b^3 c^2 (c-3 d x) (c+d x)^2-a^4 d^3 \left (3 c^2+20 c d x+15 d^2 x^2\right )+a^3 b d^2 \left (9 c^3+39 c^2 d x+11 c d^2 x^2-15 d^3 x^3\right )+a^2 b^2 c d \left (-9 c^3-9 c^2 d x+33 c d^2 x^2+31 d^3 x^3\right )}{3 a^2 c^3 (-b c+a d)^3 x \sqrt {a+b x} (c+d x)^{3/2}}+\frac {(3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{5/2} c^{7/2}} \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. \(1691\) vs.
\(2(232)=464\).
time = 0.07, size = 1692, normalized size = 6.41
method | result | size |
default | \(\text {Expression too large to display}\) | \(1692\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 819 vs.
\(2 (232) = 464\).
time = 3.55, size = 1658, normalized size = 6.28 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} - 6 \, a^{2} b^{3} c^{2} d^{4} + 12 \, a^{3} b^{2} c d^{5} - 5 \, a^{4} b d^{6}\right )} x^{4} + {\left (6 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - 16 \, a^{2} b^{3} c^{3} d^{3} + 18 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 5 \, a^{5} d^{6}\right )} x^{3} + {\left (3 \, b^{5} c^{6} + 2 \, a b^{4} c^{5} d - 14 \, a^{2} b^{3} c^{4} d^{2} + 19 \, a^{4} b c^{2} d^{4} - 10 \, a^{5} c d^{5}\right )} x^{2} + {\left (3 \, a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 12 \, a^{4} b c^{3} d^{3} - 5 \, a^{5} c^{2} d^{4}\right )} x\right )} \sqrt {a c} \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 (3 \, a^{2} b^{3} c^{6} - 9 \, a^{3} b^{2} c^{5} d + 9 \, a^{4} b c^{4} d^{2} - 3 \, a^{5} c^{3} d^{3} + {\left (9 \, a b^{4} c^{4} d^{2} - 9 \, a^{2} b^{3} c^{3} d^{3} + 31 \, a^{3} b^{2} c^{2} d^{4} - 15 \, a^{4} b c d^{5}\right )} x^{3} + {\left (18 \, a b^{4} c^{5} d - 15 \, a^{2} b^{3} c^{4} d^{2} + 33 \, a^{3} b^{2} c^{3} d^{3} + 11 \, a^{4} b c^{2} d^{4} - 15 \, a^{5} c d^{5}\right )} x^{2} + {\left (9 \, a b^{4} c^{6} - 3 \, a^{2} b^{3} c^{5} d - 9 \, a^{3} b^{2} c^{4} d^{2} + 39 \, a^{4} b c^{3} d^{3} - 20 \, a^{5} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{4} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{3} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{2} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x\right )}}, -\frac {3 \, {\left ({\left (3 \, b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} - 6 \, a^{2} b^{3} c^{2} d^{4} + 12 \, a^{3} b^{2} c d^{5} - 5 \, a^{4} b d^{6}\right )} x^{4} + {\left (6 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - 16 \, a^{2} b^{3} c^{3} d^{3} + 18 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 5 \, a^{5} d^{6}\right )} x^{3} + {\left (3 \, b^{5} c^{6} + 2 \, a b^{4} c^{5} d - 14 \, a^{2} b^{3} c^{4} d^{2} + 19 \, a^{4} b c^{2} d^{4} - 10 \, a^{5} c d^{5}\right )} x^{2} + {\left (3 \, a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d - 6 \, a^{3} b^{2} c^{4} d^{2} + 12 \, a^{4} b c^{3} d^{3} - 5 \, a^{5} c^{2} d^{4}\right )} x\right )} \sqrt {-a c} \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 (3 \, a^{2} b^{3} c^{6} - 9 \, a^{3} b^{2} c^{5} d + 9 \, a^{4} b c^{4} d^{2} - 3 \, a^{5} c^{3} d^{3} + {\left (9 \, a b^{4} c^{4} d^{2} - 9 \, a^{2} b^{3} c^{3} d^{3} + 31 \, a^{3} b^{2} c^{2} d^{4} - 15 \, a^{4} b c d^{5}\right )} x^{3} + {\left (18 \, a b^{4} c^{5} d - 15 \, a^{2} b^{3} c^{4} d^{2} + 33 \, a^{3} b^{2} c^{3} d^{3} + 11 \, a^{4} b c^{2} d^{4} - 15 \, a^{5} c d^{5}\right )} x^{2} + {\left (9 \, a b^{4} c^{6} - 3 \, a^{2} b^{3} c^{5} d - 9 \, a^{3} b^{2} c^{4} d^{2} + 39 \, a^{4} b c^{3} d^{3} - 20 \, a^{5} c^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{4} + {\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{3} + {\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{2} + {\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x\right )}}\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}{x^{2} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, 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. 1336 vs.
\(2 (232) = 464\).
time = 3.40, size = 1336, normalized size = 5.06 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (11 \, b^{6} c^{6} d^{5} {\left | b \right |} - 28 \, a b^{5} c^{5} d^{6} {\left | b \right |} + 23 \, a^{2} b^{4} c^{4} d^{7} {\left | b \right |} - 6 \, a^{3} b^{3} c^{3} d^{8} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{11} d - 5 \, a b^{6} c^{10} d^{2} + 10 \, a^{2} b^{5} c^{9} d^{3} - 10 \, a^{3} b^{4} c^{8} d^{4} + 5 \, a^{4} b^{3} c^{7} d^{5} - a^{5} b^{2} c^{6} d^{6}} + \frac {6 \, {\left (2 \, b^{7} c^{7} d^{4} {\left | b \right |} - 7 \, a b^{6} c^{6} d^{5} {\left | b \right |} + 9 \, a^{2} b^{5} c^{5} d^{6} {\left | b \right |} - 5 \, a^{3} b^{4} c^{4} d^{7} {\left | b \right |} + a^{4} b^{3} c^{3} d^{8} {\left | b \right |}\right )}}{b^{7} c^{11} d - 5 \, a b^{6} c^{10} d^{2} + 10 \, a^{2} b^{5} c^{9} d^{3} - 10 \, a^{3} b^{4} c^{8} d^{4} + 5 \, a^{4} b^{3} c^{7} d^{5} - a^{5} b^{2} c^{6} d^{6}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{9} c^{5} - 9 \, \sqrt {b d} a b^{8} c^{4} d + 12 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} - 10 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} + 5 \, \sqrt {b d} a^{4} b^{5} c d^{4} - \sqrt {b d} a^{5} b^{4} d^{5} - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{4} + 2 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c^{3} d - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{2} d^{2} + 2 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c d^{3} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c^{3} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} c^{2} d - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c d^{2} + \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} d^{3}\right )}}{{\left (a^{2} b^{2} c^{5} {\left | b \right |} - 2 \, a^{3} b c^{4} d {\left | b \right |} + a^{4} c^{3} d^{2} {\left | b \right |}\right )} {\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} - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c^{2} + 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} c d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} d^{2} + 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} c + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6}\right )}} + \frac {{\left (3 \, \sqrt {b d} b^{3} c + 5 \, \sqrt {b d} a b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{2} b c^{3} {\left | b \right |}} \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 {1}{x^2\,{\left (a+b\,x\right )}^{3/2}\,{\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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