Optimal. Leaf size=321 \[ \frac {(b d-a e)^3 (B d-A e) (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (1+m) (a+b x)}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (2+m) (a+b x)}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (3+m) (a+b x)}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (4+m) (a+b x)}+\frac {b^3 B (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (5+m) (a+b x)} \]
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Rubi [A]
time = 0.14, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {784, 78}
\begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e) (d+e x)^{m+1}}{e^5 (m+1) (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2} (-a B e-3 A b e+4 b B d)}{e^5 (m+2) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3} (-a B e-A b e+2 b B d)}{e^5 (m+3) (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4} (-3 a B e-A b e+4 b B d)}{e^5 (m+4) (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+5}}{e^5 (m+5) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 784
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^m \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^m}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{1+m}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{2+m}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{3+m}}{e^4}+\frac {b^6 B (d+e x)^{4+m}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^3 (B d-A e) (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (1+m) (a+b x)}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (2+m) (a+b x)}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (3+m) (a+b x)}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (4+m) (a+b x)}+\frac {b^3 B (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (5+m) (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 183, normalized size = 0.57 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2} (d+e x)^{1+m} \left (\frac {(b d-a e)^3 (B d-A e)}{1+m}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)}{2+m}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^2}{3+m}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^3}{4+m}+\frac {b^3 B (d+e x)^4}{5+m}\right )}{e^5 (a+b x)^3} \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. \(1285\) vs.
\(2(266)=532\).
time = 1.02, size = 1286, normalized size = 4.01
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (B \,b^{3} e^{4} m^{4} x^{4}+A \,b^{3} e^{4} m^{4} x^{3}+3 B a \,b^{2} e^{4} m^{4} x^{3}+10 B \,b^{3} e^{4} m^{3} x^{4}+3 A a \,b^{2} e^{4} m^{4} x^{2}+11 A \,b^{3} e^{4} m^{3} x^{3}+3 B \,a^{2} b \,e^{4} m^{4} x^{2}+33 B a \,b^{2} e^{4} m^{3} x^{3}-4 B \,b^{3} d \,e^{3} m^{3} x^{3}+35 B \,b^{3} e^{4} m^{2} x^{4}+3 A \,a^{2} b \,e^{4} m^{4} x +36 A a \,b^{2} e^{4} m^{3} x^{2}-3 A \,b^{3} d \,e^{3} m^{3} x^{2}+41 A \,b^{3} e^{4} m^{2} x^{3}+B \,a^{3} e^{4} m^{4} x +36 B \,a^{2} b \,e^{4} m^{3} x^{2}-9 B a \,b^{2} d \,e^{3} m^{3} x^{2}+123 B a \,b^{2} e^{4} m^{2} x^{3}-24 B \,b^{3} d \,e^{3} m^{2} x^{3}+50 B \,b^{3} e^{4} m \,x^{4}+A \,a^{3} e^{4} m^{4}+39 A \,a^{2} b \,e^{4} m^{3} x -6 A a \,b^{2} d \,e^{3} m^{3} x +147 A a \,b^{2} e^{4} m^{2} x^{2}-24 A \,b^{3} d \,e^{3} m^{2} x^{2}+61 A \,b^{3} e^{4} m \,x^{3}+13 B \,a^{3} e^{4} m^{3} x -6 B \,a^{2} b d \,e^{3} m^{3} x +147 B \,a^{2} b \,e^{4} m^{2} x^{2}-72 B a \,b^{2} d \,e^{3} m^{2} x^{2}+183 B a \,b^{2} e^{4} m \,x^{3}+12 B \,b^{3} d^{2} e^{2} m^{2} x^{2}-44 B \,b^{3} d \,e^{3} m \,x^{3}+24 B \,b^{3} x^{4} e^{4}+14 A \,a^{3} e^{4} m^{3}-3 A \,a^{2} b d \,e^{3} m^{3}+177 A \,a^{2} b \,e^{4} m^{2} x -60 A a \,b^{2} d \,e^{3} m^{2} x +234 A a \,b^{2} e^{4} m \,x^{2}+6 A \,b^{3} d^{2} e^{2} m^{2} x -51 A \,b^{3} d \,e^{3} m \,x^{2}+30 A \,b^{3} e^{4} x^{3}-B \,a^{3} d \,e^{3} m^{3}+59 B \,a^{3} e^{4} m^{2} x -60 B \,a^{2} b d \,e^{3} m^{2} x +234 B \,a^{2} b \,e^{4} m \,x^{2}+18 B a \,b^{2} d^{2} e^{2} m^{2} x -153 B a \,b^{2} d \,e^{3} m \,x^{2}+90 B a \,b^{2} e^{4} x^{3}+36 B \,b^{3} d^{2} e^{2} m \,x^{2}-24 B \,b^{3} d \,e^{3} x^{3}+71 A \,a^{3} e^{4} m^{2}-36 A \,a^{2} b d \,e^{3} m^{2}+321 A \,a^{2} b \,e^{4} m x +6 A a \,b^{2} d^{2} e^{2} m^{2}-174 A a \,b^{2} d \,e^{3} m x +120 A a \,b^{2} e^{4} x^{2}+36 A \,b^{3} d^{2} e^{2} m x -30 A \,b^{3} d \,e^{3} x^{2}-12 B \,a^{3} d \,e^{3} m^{2}+107 B \,a^{3} e^{4} m x +6 B \,a^{2} b \,d^{2} e^{2} m^{2}-174 B \,a^{2} b d \,e^{3} m x +120 B \,a^{2} b \,e^{4} x^{2}+108 B a \,b^{2} d^{2} e^{2} m x -90 B a \,b^{2} d \,e^{3} x^{2}-24 B \,b^{3} d^{3} e m x +24 B \,b^{3} d^{2} e^{2} x^{2}+154 A \,a^{3} e^{4} m -141 A \,a^{2} b d \,e^{3} m +180 A \,a^{2} b \,e^{4} x +54 A a \,b^{2} d^{2} e^{2} m -120 A a \,b^{2} d \,e^{3} x -6 A \,b^{3} d^{3} e m +30 A \,b^{3} d^{2} e^{2} x -47 B \,a^{3} d \,e^{3} m +60 B \,a^{3} e^{4} x +54 B \,a^{2} b \,d^{2} e^{2} m -120 B \,a^{2} b d \,e^{3} x -18 B a \,b^{2} d^{3} e m +90 B a \,b^{2} d^{2} e^{2} x -24 B \,b^{3} d^{3} e x +120 A \,a^{3} e^{4}-180 A \,a^{2} b d \,e^{3}+120 A a \,b^{2} d^{2} e^{2}-30 A \,b^{3} d^{3} e -60 B \,a^{3} d \,e^{3}+120 B \,a^{2} b \,d^{2} e^{2}-90 B a \,b^{2} d^{3} e +24 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{\left (b x +a \right )^{3} e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(1286\) |
risch | \(\text {Expression too large to display}\) | \(1728\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 738 vs.
\(2 (276) = 552\).
time = 0.30, size = 738, normalized size = 2.30 \begin {gather*} \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3} x^{4} e^{4} + 6 \, a b^{2} d^{3} {\left (m + 4\right )} e - 6 \, b^{3} d^{4} - 3 \, {\left (m^{2} + 7 \, m + 12\right )} a^{2} b d^{2} e^{2} + {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} d e^{3} + {\left ({\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d e^{3} + 3 \, {\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a b^{2} e^{4}\right )} x^{3} - 3 \, {\left ({\left (m^{2} + m\right )} b^{3} d^{2} e^{2} - {\left (m^{3} + 5 \, m^{2} + 4 \, m\right )} a b^{2} d e^{3} - {\left (m^{3} + 8 \, m^{2} + 19 \, m + 12\right )} a^{2} b e^{4}\right )} x^{2} + {\left (6 \, b^{3} d^{3} m e - 6 \, {\left (m^{2} + 4 \, m\right )} a b^{2} d^{2} e^{2} + 3 \, {\left (m^{3} + 7 \, m^{2} + 12 \, m\right )} a^{2} b d e^{3} + {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} e^{4}\right )} x\right )} A e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{3} x^{5} e^{5} - 18 \, a b^{2} d^{4} {\left (m + 5\right )} e + 24 \, b^{3} d^{5} + 6 \, {\left (m^{2} + 9 \, m + 20\right )} a^{2} b d^{3} e^{2} - {\left (m^{3} + 12 \, m^{2} + 47 \, m + 60\right )} a^{3} d^{2} e^{3} + {\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{3} d e^{4} + 3 \, {\left (m^{4} + 11 \, m^{3} + 41 \, m^{2} + 61 \, m + 30\right )} a b^{2} e^{5}\right )} x^{4} - {\left (4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d^{2} e^{3} - 3 \, {\left (m^{4} + 8 \, m^{3} + 17 \, m^{2} + 10 \, m\right )} a b^{2} d e^{4} - 3 \, {\left (m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40\right )} a^{2} b e^{5}\right )} x^{3} + {\left (12 \, {\left (m^{2} + m\right )} b^{3} d^{3} e^{2} - 9 \, {\left (m^{3} + 6 \, m^{2} + 5 \, m\right )} a b^{2} d^{2} e^{3} + 3 \, {\left (m^{4} + 10 \, m^{3} + 29 \, m^{2} + 20 \, m\right )} a^{2} b d e^{4} + {\left (m^{4} + 13 \, m^{3} + 59 \, m^{2} + 107 \, m + 60\right )} a^{3} e^{5}\right )} x^{2} - {\left (24 \, b^{3} d^{4} m e - 18 \, {\left (m^{2} + 5 \, m\right )} a b^{2} d^{3} e^{2} + 6 \, {\left (m^{3} + 9 \, m^{2} + 20 \, m\right )} a^{2} b d^{2} e^{3} - {\left (m^{4} + 12 \, m^{3} + 47 \, m^{2} + 60 \, m\right )} a^{3} d e^{4}\right )} x\right )} B e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \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. 1156 vs.
\(2 (276) = 552\).
time = 1.97, size = 1156, normalized size = 3.60 \begin {gather*} \frac {{\left (24 \, B b^{3} d^{5} + {\left ({\left (B b^{3} m^{4} + 10 \, B b^{3} m^{3} + 35 \, B b^{3} m^{2} + 50 \, B b^{3} m + 24 \, B b^{3}\right )} x^{5} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 90 \, B a b^{2} + 30 \, A b^{3} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 41 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 61 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{4} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 40 \, B a^{2} b + 40 \, A a b^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 49 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 78 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 60 \, B a^{3} + 180 \, A a^{2} b + 13 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 59 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 107 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{2} + {\left (A a^{3} m^{4} + 14 \, A a^{3} m^{3} + 71 \, A a^{3} m^{2} + 154 \, A a^{3} m + 120 \, A a^{3}\right )} x\right )} e^{5} + {\left (A a^{3} d m^{4} + 14 \, A a^{3} d m^{3} + 71 \, A a^{3} d m^{2} + 154 \, A a^{3} d m + 120 \, A a^{3} d + {\left (B b^{3} d m^{4} + 6 \, B b^{3} d m^{3} + 11 \, B b^{3} d m^{2} + 6 \, B b^{3} d m\right )} x^{4} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d m^{4} + 8 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d m^{3} + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d m^{2} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d m\right )} x^{3} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} d m^{4} + 10 \, {\left (B a^{2} b + A a b^{2}\right )} d m^{3} + 29 \, {\left (B a^{2} b + A a b^{2}\right )} d m^{2} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} d m\right )} x^{2} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} d m^{4} + 12 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d m^{3} + 47 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d m^{2} + 60 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d m\right )} x\right )} e^{4} - {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} m^{3} + 12 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} m^{2} + 47 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} m + 4 \, {\left (B b^{3} d^{2} m^{3} + 3 \, B b^{3} d^{2} m^{2} + 2 \, B b^{3} d^{2} m\right )} x^{3} + 60 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} + 3 \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} m^{3} + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} m^{2} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} m\right )} x^{2} + 6 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} d^{2} m^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} m^{2} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} m\right )} x\right )} e^{3} + 6 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} d^{3} m^{2} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} m + 20 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} + 2 \, {\left (B b^{3} d^{3} m^{2} + B b^{3} d^{3} m\right )} x^{2} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} m^{2} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} m\right )} x\right )} e^{2} - 6 \, {\left (4 \, B b^{3} d^{4} m x + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} m + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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. 3208 vs.
\(2 (276) = 552\).
time = 1.11, size = 3208, normalized size = 9.99 \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 \left (A+B\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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