Optimal. Leaf size=437 \[ -\frac {(b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^5 d^4}+\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {(b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{11/2} d^{9/2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65,
223, 212} \begin {gather*} \frac {\left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{11/2} d^{9/2}}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{384 b^5 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^4}{1024 b^5 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{1536 b^5 d^3}+\frac {(a+b x)^{5/2} (c+d x)^{3/2} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{120 b^3 d^2}-\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rubi steps
\begin {align*} \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx &=\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\int (a+b x)^{3/2} (c+d x)^{5/2} \left (-a c-\frac {1}{2} (7 b c+9 a d) x\right ) \, dx}{7 b d}\\ &=-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \int (a+b x)^{3/2} (c+d x)^{5/2} \, dx}{24 b^2 d^2}\\ &=\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left ((b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right )\right ) \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx}{48 b^3 d^2}\\ &=\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left ((b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right )\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{128 b^4 d^2}\\ &=\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left ((b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{768 b^5 d^2}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{1024 b^5 d^3}\\ &=-\frac {(b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^5 d^4}+\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left ((b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2048 b^5 d^4}\\ &=-\frac {(b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^5 d^4}+\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left ((b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 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 )}{1024 b^6 d^4}\\ &=-\frac {(b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^5 d^4}+\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {\left ((b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 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 )}{1024 b^6 d^4}\\ &=-\frac {(b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^5 d^4}+\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {(b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{11/2} d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 382, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (945 a^6 d^6-210 a^5 b d^5 (16 c+3 d x)+7 a^4 b^2 d^4 \left (527 c^2+314 c d x+72 d^2 x^2\right )-4 a^3 b^3 d^3 \left (150 c^3+583 c^2 d x+436 c d^2 x^2+108 d^3 x^3\right )+3 a^2 b^4 d^2 \left (-175 c^4+100 c^3 d x+608 c^2 d^2 x^2+496 c d^3 x^3+128 d^4 x^4\right )+10 a b^5 d \left (140 c^5-91 c^4 d x+72 c^3 d^2 x^2+3352 c^2 d^3 x^3+4864 c d^4 x^4+1920 d^5 x^5\right )-5 b^6 \left (105 c^6-70 c^5 d x+56 c^4 d^2 x^2-48 c^3 d^3 x^3-4736 c^2 d^4 x^4-7424 c d^5 x^5-3072 d^6 x^6\right )\right )}{107520 b^5 d^4}+\frac {(b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{1024 b^{11/2} 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. \(1320\) vs.
\(2(381)=762\).
time = 0.07, size = 1321, normalized size = 3.02
method | result | size |
default | \(\text {Expression too large to display}\) | \(1321\) |
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 = 1.11, size = 1110, normalized size = 2.54 \begin {gather*} \left [-\frac {105 \, {\left (5 \, b^{7} c^{7} - 15 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 15 \, a^{4} b^{3} c^{3} d^{4} - 45 \, a^{5} b^{2} c^{2} d^{5} + 35 \, a^{6} b c d^{6} - 9 \, a^{7} d^{7}\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 (15360 \, b^{7} d^{7} x^{6} - 525 \, b^{7} c^{6} d + 1400 \, a b^{6} c^{5} d^{2} - 525 \, a^{2} b^{5} c^{4} d^{3} - 600 \, a^{3} b^{4} c^{3} d^{4} + 3689 \, a^{4} b^{3} c^{2} d^{5} - 3360 \, a^{5} b^{2} c d^{6} + 945 \, a^{6} b d^{7} + 1280 \, {\left (29 \, b^{7} c d^{6} + 15 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (185 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 3 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (15 \, b^{7} c^{3} d^{4} + 2095 \, a b^{6} c^{2} d^{5} + 93 \, a^{2} b^{5} c d^{6} - 27 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (35 \, b^{7} c^{4} d^{3} - 90 \, a b^{6} c^{3} d^{4} - 228 \, a^{2} b^{5} c^{2} d^{5} + 218 \, a^{3} b^{4} c d^{6} - 63 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (175 \, b^{7} c^{5} d^{2} - 455 \, a b^{6} c^{4} d^{3} + 150 \, a^{2} b^{5} c^{3} d^{4} - 1166 \, a^{3} b^{4} c^{2} d^{5} + 1099 \, a^{4} b^{3} c d^{6} - 315 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{430080 \, b^{6} d^{5}}, -\frac {105 \, {\left (5 \, b^{7} c^{7} - 15 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 15 \, a^{4} b^{3} c^{3} d^{4} - 45 \, a^{5} b^{2} c^{2} d^{5} + 35 \, a^{6} b c d^{6} - 9 \, a^{7} d^{7}\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 (15360 \, b^{7} d^{7} x^{6} - 525 \, b^{7} c^{6} d + 1400 \, a b^{6} c^{5} d^{2} - 525 \, a^{2} b^{5} c^{4} d^{3} - 600 \, a^{3} b^{4} c^{3} d^{4} + 3689 \, a^{4} b^{3} c^{2} d^{5} - 3360 \, a^{5} b^{2} c d^{6} + 945 \, a^{6} b d^{7} + 1280 \, {\left (29 \, b^{7} c d^{6} + 15 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (185 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 3 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (15 \, b^{7} c^{3} d^{4} + 2095 \, a b^{6} c^{2} d^{5} + 93 \, a^{2} b^{5} c d^{6} - 27 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (35 \, b^{7} c^{4} d^{3} - 90 \, a b^{6} c^{3} d^{4} - 228 \, a^{2} b^{5} c^{2} d^{5} + 218 \, a^{3} b^{4} c d^{6} - 63 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (175 \, b^{7} c^{5} d^{2} - 455 \, a b^{6} c^{4} d^{3} + 150 \, a^{2} b^{5} c^{3} d^{4} - 1166 \, a^{3} b^{4} c^{2} d^{5} + 1099 \, a^{4} b^{3} c d^{6} - 315 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{215040 \, b^{6} d^{5}}\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 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. 3513 vs.
\(2 (381) = 762\).
time = 2.13, size = 3513, normalized size = 8.04 \begin {gather*} \text {Too large to display} \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 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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