Optimal. Leaf size=251 \[ \frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^2 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^3 c^3-31 a b^2 c^2 d+9 a^2 b c d^2-9 a^3 d^3\right )+d (b c-a d) \left (5 b^2 c^2-6 a b c d+9 a^2 d^2\right ) x\right )}{3 b^2 d^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{7/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 155, 148,
65, 223, 212} \begin {gather*} \frac {\sqrt {a+b x} \left (d x (b c-a d) \left (9 a^2 d^2-6 a b c d+5 b^2 c^2\right )+c \left (-9 a^3 d^3+9 a^2 b c d^2-31 a b^2 c^2 d+15 b^3 c^3\right )\right )}{3 b^2 d^3 \sqrt {c+d x} (b c-a d)^3}-\frac {(3 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{7/2}}+\frac {2 a x^3}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (3 a d+b c)}{3 b d (c+d x)^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 148
Rule 155
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \int \frac {x^2 \left (3 a c+\frac {1}{2} (-b c+3 a d) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b (b c-a d)}\\ &=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^2 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 \int \frac {x \left (a c (b c+3 a d)+\frac {1}{4} \left (5 b^2 c^2-6 a b c d+9 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 b d (b c-a d)^2}\\ &=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^2 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^3 c^3-31 a b^2 c^2 d+9 a^2 b c d^2-9 a^3 d^3\right )+d (b c-a d) \left (5 b^2 c^2-6 a b c d+9 a^2 d^2\right ) x\right )}{3 b^2 d^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(5 b c+3 a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2 d^3}\\ &=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^2 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^3 c^3-31 a b^2 c^2 d+9 a^2 b c d^2-9 a^3 d^3\right )+d (b c-a d) \left (5 b^2 c^2-6 a b c d+9 a^2 d^2\right ) x\right )}{3 b^2 d^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(5 b c+3 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3 d^3}\\ &=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^2 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^3 c^3-31 a b^2 c^2 d+9 a^2 b c d^2-9 a^3 d^3\right )+d (b c-a d) \left (5 b^2 c^2-6 a b c d+9 a^2 d^2\right ) x\right )}{3 b^2 d^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(5 b c+3 a d) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^3 d^3}\\ &=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^2 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^3 c^3-31 a b^2 c^2 d+9 a^2 b c d^2-9 a^3 d^3\right )+d (b c-a d) \left (5 b^2 c^2-6 a b c d+9 a^2 d^2\right ) x\right )}{3 b^2 d^3 (b c-a d)^3 \sqrt {c+d x}}-\frac {(5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 234, normalized size = 0.93 \begin {gather*} \frac {9 a^4 d^3 (c+d x)^2+3 a^3 b d^2 (-3 c+d x) (c+d x)^2-b^4 c^3 x \left (15 c^2+20 c d x+3 d^2 x^2\right )+a^2 b^2 c d \left (31 c^3+33 c^2 d x-9 c d^2 x^2-9 d^3 x^3\right )+a b^3 c^2 \left (-15 c^3+11 c^2 d x+39 c d^2 x^2+9 d^3 x^3\right )}{3 b^2 d^3 (-b c+a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {(5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2} d^{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. \(1713\) vs.
\(2(223)=446\).
time = 0.08, size = 1714, normalized size = 6.83
method | result | size |
default | \(\text {Expression too large to display}\) | \(1714\) |
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. 807 vs.
\(2 (223) = 446\).
time = 2.11, size = 1628, normalized size = 6.49 \begin {gather*} \left [\frac {3 \, {\left (5 \, a b^{4} c^{6} - 12 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} + 4 \, a^{4} b c^{3} d^{3} - 3 \, a^{5} c^{2} d^{4} + {\left (5 \, b^{5} c^{4} d^{2} - 12 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} + 4 \, a^{3} b^{2} c d^{5} - 3 \, a^{4} b d^{6}\right )} x^{3} + {\left (10 \, b^{5} c^{5} d - 19 \, a b^{4} c^{4} d^{2} + 14 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} - 3 \, a^{5} d^{6}\right )} x^{2} + {\left (5 \, b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 18 \, a^{2} b^{3} c^{4} d^{2} + 16 \, a^{3} b^{2} c^{3} d^{3} + 5 \, a^{4} b c^{2} d^{4} - 6 \, a^{5} c d^{5}\right )} x\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 (15 \, a b^{4} c^{5} d - 31 \, a^{2} b^{3} c^{4} d^{2} + 9 \, a^{3} b^{2} c^{3} d^{3} - 9 \, a^{4} b c^{2} d^{4} + 3 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )} x^{3} + {\left (20 \, b^{5} c^{4} d^{2} - 39 \, a b^{4} c^{3} d^{3} + 9 \, a^{2} b^{3} c^{2} d^{4} + 3 \, a^{3} b^{2} c d^{5} - 9 \, a^{4} b d^{6}\right )} x^{2} + {\left (15 \, b^{5} c^{5} d - 11 \, a b^{4} c^{4} d^{2} - 33 \, a^{2} b^{3} c^{3} d^{3} + 15 \, a^{3} b^{2} c^{2} d^{4} - 18 \, a^{4} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (a b^{6} c^{5} d^{4} - 3 \, a^{2} b^{5} c^{4} d^{5} + 3 \, a^{3} b^{4} c^{3} d^{6} - a^{4} b^{3} c^{2} d^{7} + {\left (b^{7} c^{3} d^{6} - 3 \, a b^{6} c^{2} d^{7} + 3 \, a^{2} b^{5} c d^{8} - a^{3} b^{4} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{4} d^{5} - 5 \, a b^{6} c^{3} d^{6} + 3 \, a^{2} b^{5} c^{2} d^{7} + a^{3} b^{4} c d^{8} - a^{4} b^{3} d^{9}\right )} x^{2} + {\left (b^{7} c^{5} d^{4} - a b^{6} c^{4} d^{5} - 3 \, a^{2} b^{5} c^{3} d^{6} + 5 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8}\right )} x\right )}}, \frac {3 \, {\left (5 \, a b^{4} c^{6} - 12 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} + 4 \, a^{4} b c^{3} d^{3} - 3 \, a^{5} c^{2} d^{4} + {\left (5 \, b^{5} c^{4} d^{2} - 12 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} + 4 \, a^{3} b^{2} c d^{5} - 3 \, a^{4} b d^{6}\right )} x^{3} + {\left (10 \, b^{5} c^{5} d - 19 \, a b^{4} c^{4} d^{2} + 14 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} - 3 \, a^{5} d^{6}\right )} x^{2} + {\left (5 \, b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 18 \, a^{2} b^{3} c^{4} d^{2} + 16 \, a^{3} b^{2} c^{3} d^{3} + 5 \, a^{4} b c^{2} d^{4} - 6 \, a^{5} c d^{5}\right )} x\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 (15 \, a b^{4} c^{5} d - 31 \, a^{2} b^{3} c^{4} d^{2} + 9 \, a^{3} b^{2} c^{3} d^{3} - 9 \, a^{4} b c^{2} d^{4} + 3 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )} x^{3} + {\left (20 \, b^{5} c^{4} d^{2} - 39 \, a b^{4} c^{3} d^{3} + 9 \, a^{2} b^{3} c^{2} d^{4} + 3 \, a^{3} b^{2} c d^{5} - 9 \, a^{4} b d^{6}\right )} x^{2} + {\left (15 \, b^{5} c^{5} d - 11 \, a b^{4} c^{4} d^{2} - 33 \, a^{2} b^{3} c^{3} d^{3} + 15 \, a^{3} b^{2} c^{2} d^{4} - 18 \, a^{4} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a b^{6} c^{5} d^{4} - 3 \, a^{2} b^{5} c^{4} d^{5} + 3 \, a^{3} b^{4} c^{3} d^{6} - a^{4} b^{3} c^{2} d^{7} + {\left (b^{7} c^{3} d^{6} - 3 \, a b^{6} c^{2} d^{7} + 3 \, a^{2} b^{5} c d^{8} - a^{3} b^{4} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{4} d^{5} - 5 \, a b^{6} c^{3} d^{6} + 3 \, a^{2} b^{5} c^{2} d^{7} + a^{3} b^{4} c d^{8} - a^{4} b^{3} d^{9}\right )} x^{2} + {\left (b^{7} c^{5} d^{4} - a b^{6} c^{4} d^{5} - 3 \, a^{2} b^{5} c^{3} d^{6} + 5 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8}\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 {x^{4}}{\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. 753 vs.
\(2 (223) = 446\).
time = 1.42, size = 753, normalized size = 3.00 \begin {gather*} -\frac {4 \, \sqrt {b d} a^{4}}{{\left (b^{3} c^{2} {\left | b \right |} - 2 \, a b^{2} c d {\left | b \right |} + a^{2} b d^{2} {\left | b \right |}\right )} {\left (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}\right )}} + \frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{8} c^{5} d^{4} {\left | b \right |} - 5 \, a b^{7} c^{4} d^{5} {\left | b \right |} + 10 \, a^{2} b^{6} c^{3} d^{6} {\left | b \right |} - 10 \, a^{3} b^{5} c^{2} d^{7} {\left | b \right |} + 5 \, a^{4} b^{4} c d^{8} {\left | b \right |} - a^{5} b^{3} d^{9} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}} + \frac {2 \, {\left (10 \, b^{9} c^{6} d^{3} {\left | b \right |} - 44 \, a b^{8} c^{5} d^{4} {\left | b \right |} + 76 \, a^{2} b^{7} c^{4} d^{5} {\left | b \right |} - 72 \, a^{3} b^{6} c^{3} d^{6} {\left | b \right |} + 45 \, a^{4} b^{5} c^{2} d^{7} {\left | b \right |} - 18 \, a^{5} b^{4} c d^{8} {\left | b \right |} + 3 \, a^{6} b^{3} d^{9} {\left | b \right |}\right )}}{b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}}\right )} + \frac {3 \, {\left (5 \, b^{10} c^{7} d^{2} {\left | b \right |} - 27 \, a b^{9} c^{6} d^{3} {\left | b \right |} + 57 \, a^{2} b^{8} c^{5} d^{4} {\left | b \right |} - 63 \, a^{3} b^{7} c^{4} d^{5} {\left | b \right |} + 43 \, a^{4} b^{6} c^{3} d^{6} {\left | b \right |} - 21 \, a^{5} b^{5} c^{2} d^{7} {\left | b \right |} + 7 \, a^{6} b^{4} c d^{8} {\left | b \right |} - a^{7} b^{3} d^{9} {\left | b \right |}\right )}}{b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, \sqrt {b d} b c + 3 \, \sqrt {b d} a 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 )}^{2}\right )}{2 \, b^{2} d^{4} {\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 {x^4}{{\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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