Optimal. Leaf size=351 \[ \frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 155, 152,
65, 223, 212} \begin {gather*} -\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{3 b d^2 \sqrt {c+d x} (b c-a d)^3}+\frac {5 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {2 c x^3 \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 152
Rule 155
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \int \frac {x^3 \left (4 a c+\frac {1}{2} (-b c+5 a d) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b (b c-a d)}\\ &=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}+\frac {4 \int \frac {x^2 \left (\frac {3}{2} a c (b c+3 a d)+\frac {1}{4} \left (7 b^2 c^2-6 a b c d+15 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^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {8 \int \frac {x \left (-\frac {1}{2} a c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right )+\frac {1}{8} \left (-35 b^3 c^3+61 a b^2 c^2 d-9 a^2 b c d^2+15 a^3 d^3\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 b d^2 (b c-a d)^3}\\ &=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {\left (5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^3 d^4}\\ &=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {\left (5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\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 )}{4 b^4 d^4}\\ &=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {\left (5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\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 )}{4 b^4 d^4}\\ &=\frac {2 a x^4}{b (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c (b c+3 a d) x^3 \sqrt {a+b x}}{3 b d (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 c \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b d^2 (b c-a d)^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4-2 b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{12 b^3 d^4 (b c-a d)^3}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 320, normalized size = 0.91 \begin {gather*} \frac {45 a^5 d^4 (c+d x)^2+15 a^4 b d^3 (-2 c+d x) (c+d x)^2-6 a^3 b^2 d^2 (c+d x)^2 \left (6 c^2+2 c d x+d^2 x^2\right )-b^5 c^3 x \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )+a b^4 c^2 \left (-105 c^4+50 c^3 d x+237 c^2 d^2 x^2+48 c d^3 x^3-18 d^4 x^4\right )+2 a^2 b^3 c d \left (95 c^4+111 c^3 d x-6 c^2 d^2 x^2-9 c d^3 x^3+9 d^4 x^4\right )}{12 b^3 d^4 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{7/2} d^{9/2}} \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. \(2227\) vs.
\(2(315)=630\).
time = 0.08, size = 2228, normalized size = 6.35
method | result | size |
default | \(\text {Expression too large to display}\) | \(2228\) |
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. 973 vs.
\(2 (315) = 630\).
time = 3.04, size = 1960, normalized size = 5.58 \begin {gather*} \left [\frac {15 \, {\left (7 \, a b^{5} c^{7} - 15 \, a^{2} b^{4} c^{6} d + 6 \, a^{3} b^{3} c^{5} d^{2} + 2 \, a^{4} b^{2} c^{4} d^{3} + 3 \, a^{5} b c^{3} d^{4} - 3 \, a^{6} c^{2} d^{5} + {\left (7 \, b^{6} c^{5} d^{2} - 15 \, a b^{5} c^{4} d^{3} + 6 \, a^{2} b^{4} c^{3} d^{4} + 2 \, a^{3} b^{3} c^{2} d^{5} + 3 \, a^{4} b^{2} c d^{6} - 3 \, a^{5} b d^{7}\right )} x^{3} + {\left (14 \, b^{6} c^{6} d - 23 \, a b^{5} c^{5} d^{2} - 3 \, a^{2} b^{4} c^{4} d^{3} + 10 \, a^{3} b^{3} c^{3} d^{4} + 8 \, a^{4} b^{2} c^{2} d^{5} - 3 \, a^{5} b c d^{6} - 3 \, a^{6} d^{7}\right )} x^{2} + {\left (7 \, b^{6} c^{7} - a b^{5} c^{6} d - 24 \, a^{2} b^{4} c^{5} d^{2} + 14 \, a^{3} b^{3} c^{4} d^{3} + 7 \, a^{4} b^{2} c^{3} d^{4} + 3 \, a^{5} b c^{2} d^{5} - 6 \, a^{6} c d^{6}\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 (105 \, a b^{5} c^{6} d - 190 \, a^{2} b^{4} c^{5} d^{2} + 36 \, a^{3} b^{3} c^{4} d^{3} + 30 \, a^{4} b^{2} c^{3} d^{4} - 45 \, a^{5} b c^{2} d^{5} - 6 \, {\left (b^{6} c^{3} d^{4} - 3 \, a b^{5} c^{2} d^{5} + 3 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )} x^{4} + 3 \, {\left (7 \, b^{6} c^{4} d^{3} - 16 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} + 8 \, a^{3} b^{3} c d^{6} - 5 \, a^{4} b^{2} d^{7}\right )} x^{3} + {\left (140 \, b^{6} c^{5} d^{2} - 237 \, a b^{5} c^{4} d^{3} + 12 \, a^{2} b^{4} c^{3} d^{4} + 66 \, a^{3} b^{3} c^{2} d^{5} - 45 \, a^{5} b d^{7}\right )} x^{2} + {\left (105 \, b^{6} c^{6} d - 50 \, a b^{5} c^{5} d^{2} - 222 \, a^{2} b^{4} c^{4} d^{3} + 84 \, a^{3} b^{3} c^{3} d^{4} + 45 \, a^{4} b^{2} c^{2} d^{5} - 90 \, a^{5} b c d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a b^{7} c^{5} d^{5} - 3 \, a^{2} b^{6} c^{4} d^{6} + 3 \, a^{3} b^{5} c^{3} d^{7} - a^{4} b^{4} c^{2} d^{8} + {\left (b^{8} c^{3} d^{7} - 3 \, a b^{7} c^{2} d^{8} + 3 \, a^{2} b^{6} c d^{9} - a^{3} b^{5} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{4} d^{6} - 5 \, a b^{7} c^{3} d^{7} + 3 \, a^{2} b^{6} c^{2} d^{8} + a^{3} b^{5} c d^{9} - a^{4} b^{4} d^{10}\right )} x^{2} + {\left (b^{8} c^{5} d^{5} - a b^{7} c^{4} d^{6} - 3 \, a^{2} b^{6} c^{3} d^{7} + 5 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9}\right )} x\right )}}, -\frac {15 \, {\left (7 \, a b^{5} c^{7} - 15 \, a^{2} b^{4} c^{6} d + 6 \, a^{3} b^{3} c^{5} d^{2} + 2 \, a^{4} b^{2} c^{4} d^{3} + 3 \, a^{5} b c^{3} d^{4} - 3 \, a^{6} c^{2} d^{5} + {\left (7 \, b^{6} c^{5} d^{2} - 15 \, a b^{5} c^{4} d^{3} + 6 \, a^{2} b^{4} c^{3} d^{4} + 2 \, a^{3} b^{3} c^{2} d^{5} + 3 \, a^{4} b^{2} c d^{6} - 3 \, a^{5} b d^{7}\right )} x^{3} + {\left (14 \, b^{6} c^{6} d - 23 \, a b^{5} c^{5} d^{2} - 3 \, a^{2} b^{4} c^{4} d^{3} + 10 \, a^{3} b^{3} c^{3} d^{4} + 8 \, a^{4} b^{2} c^{2} d^{5} - 3 \, a^{5} b c d^{6} - 3 \, a^{6} d^{7}\right )} x^{2} + {\left (7 \, b^{6} c^{7} - a b^{5} c^{6} d - 24 \, a^{2} b^{4} c^{5} d^{2} + 14 \, a^{3} b^{3} c^{4} d^{3} + 7 \, a^{4} b^{2} c^{3} d^{4} + 3 \, a^{5} b c^{2} d^{5} - 6 \, a^{6} c d^{6}\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 (105 \, a b^{5} c^{6} d - 190 \, a^{2} b^{4} c^{5} d^{2} + 36 \, a^{3} b^{3} c^{4} d^{3} + 30 \, a^{4} b^{2} c^{3} d^{4} - 45 \, a^{5} b c^{2} d^{5} - 6 \, {\left (b^{6} c^{3} d^{4} - 3 \, a b^{5} c^{2} d^{5} + 3 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )} x^{4} + 3 \, {\left (7 \, b^{6} c^{4} d^{3} - 16 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} + 8 \, a^{3} b^{3} c d^{6} - 5 \, a^{4} b^{2} d^{7}\right )} x^{3} + {\left (140 \, b^{6} c^{5} d^{2} - 237 \, a b^{5} c^{4} d^{3} + 12 \, a^{2} b^{4} c^{3} d^{4} + 66 \, a^{3} b^{3} c^{2} d^{5} - 45 \, a^{5} b d^{7}\right )} x^{2} + {\left (105 \, b^{6} c^{6} d - 50 \, a b^{5} c^{5} d^{2} - 222 \, a^{2} b^{4} c^{4} d^{3} + 84 \, a^{3} b^{3} c^{3} d^{4} + 45 \, a^{4} b^{2} c^{2} d^{5} - 90 \, a^{5} b c d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a b^{7} c^{5} d^{5} - 3 \, a^{2} b^{6} c^{4} d^{6} + 3 \, a^{3} b^{5} c^{3} d^{7} - a^{4} b^{4} c^{2} d^{8} + {\left (b^{8} c^{3} d^{7} - 3 \, a b^{7} c^{2} d^{8} + 3 \, a^{2} b^{6} c d^{9} - a^{3} b^{5} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{4} d^{6} - 5 \, a b^{7} c^{3} d^{7} + 3 \, a^{2} b^{6} c^{2} d^{8} + a^{3} b^{5} c d^{9} - a^{4} b^{4} d^{10}\right )} x^{2} + {\left (b^{8} c^{5} d^{5} - a b^{7} c^{4} d^{6} - 3 \, a^{2} b^{6} c^{3} d^{7} + 5 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9}\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^{5}}{\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. 983 vs.
\(2 (315) = 630\).
time = 1.21, size = 983, normalized size = 2.80 \begin {gather*} \frac {4 \, \sqrt {b d} a^{5}}{{\left (b^{4} c^{2} {\left | b \right |} - 2 \, a b^{3} c d {\left | b \right |} + a^{2} b^{2} 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 (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{14} c^{5} d^{6} - 5 \, a b^{13} c^{4} d^{7} + 10 \, a^{2} b^{12} c^{3} d^{8} - 10 \, a^{3} b^{11} c^{2} d^{9} + 5 \, a^{4} b^{10} c d^{10} - a^{5} b^{9} d^{11}\right )} {\left (b x + a\right )}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}} - \frac {7 \, b^{15} c^{6} d^{5} - 22 \, a b^{14} c^{5} d^{6} + 5 \, a^{2} b^{13} c^{4} d^{7} + 60 \, a^{3} b^{12} c^{3} d^{8} - 95 \, a^{4} b^{11} c^{2} d^{9} + 58 \, a^{5} b^{10} c d^{10} - 13 \, a^{6} b^{9} d^{11}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}}\right )} - \frac {20 \, {\left (7 \, b^{16} c^{7} d^{4} - 29 \, a b^{15} c^{6} d^{5} + 43 \, a^{2} b^{14} c^{5} d^{6} - 21 \, a^{3} b^{13} c^{4} d^{7} - 15 \, a^{4} b^{12} c^{3} d^{8} + 27 \, a^{5} b^{11} c^{2} d^{9} - 15 \, a^{6} b^{10} c d^{10} + 3 \, a^{7} b^{9} d^{11}\right )}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{17} c^{8} d^{3} - 180 \, a b^{16} c^{7} d^{4} + 360 \, a^{2} b^{15} c^{6} d^{5} - 340 \, a^{3} b^{14} c^{5} d^{6} + 110 \, a^{4} b^{13} c^{4} d^{7} + 84 \, a^{5} b^{12} c^{3} d^{8} - 112 \, a^{6} b^{11} c^{2} d^{9} + 52 \, a^{7} b^{10} c d^{10} - 9 \, a^{8} b^{9} d^{11}\right )}}{b^{15} c^{5} d^{7} {\left | b \right |} - 5 \, a b^{14} c^{4} d^{8} {\left | b \right |} + 10 \, a^{2} b^{13} c^{3} d^{9} {\left | b \right |} - 10 \, a^{3} b^{12} c^{2} d^{10} {\left | b \right |} + 5 \, a^{4} b^{11} c d^{11} {\left | b \right |} - a^{5} b^{10} d^{12} {\left | b \right |}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, \sqrt {b d} b^{2} c^{2} + 6 \, \sqrt {b d} a b c d + 3 \, \sqrt {b d} a^{2} 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 )}^{2}\right )}{8 \, b^{3} d^{5} {\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^5}{{\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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