Optimal. Leaf size=248 \[ \frac {(a+b x) (d+e x)^3 (A b-a B)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B) (b d-a e)^3 \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x (a+b x) (A b-a B) (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2 (A b-a B) (b d-a e)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) (d+e x)^4}{4 b e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 248, 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 = {770, 77} \begin {gather*} \frac {(a+b x) (d+e x)^3 (A b-a B)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2 (A b-a B) (b d-a e)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x (a+b x) (A b-a B) (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B) (b d-a e)^3 \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) (d+e x)^4}{4 b e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(A+B x) (d+e x)^3}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {(A b-a B) e (b d-a e)^2}{b^5}+\frac {(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac {(A b-a B) e (b d-a e) (d+e x)}{b^4}+\frac {(A b-a B) e (d+e x)^2}{b^3}+\frac {B (d+e x)^3}{b^2}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) e (b d-a e)^2 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (b d-a e) (a+b x) (d+e x)^2}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x) (d+e x)^3}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B (a+b x) (d+e x)^4}{4 b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (b d-a e)^3 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 185, normalized size = 0.75 \begin {gather*} \frac {(a+b x) \left (b x \left (-12 a^3 B e^3+6 a^2 b e^2 (2 A e+6 B d+B e x)-2 a b^2 e \left (3 A e (6 d+e x)+B \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+b^3 \left (2 A e \left (18 d^2+9 d e x+2 e^2 x^2\right )+3 B \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )\right )+12 (A b-a B) (b d-a e)^3 \log (a+b x)\right )}{12 b^5 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.83, size = 959, normalized size = 3.87 \begin {gather*} \frac {\sqrt {a^2+2 b x a+b^2 x^2} \left (12 B d^3 b^3+3 B e^3 x^3 b^3+4 A e^3 x^2 b^3+12 B d e^2 x^2 b^3+36 A d^2 e b^3+18 A d e^2 x b^3+18 B d^2 e x b^3-54 a A d e^2 b^2-7 a B e^3 x^2 b^2-54 a B d^2 e b^2-10 a A e^3 x b^2-30 a B d e^2 x b^2+22 a^2 A e^3 b+66 a^2 B d e^2 b+13 a^2 B e^3 x b-25 a^3 B e^3\right )}{24 b^5}+\frac {-3 b^3 B e^3 x^4-4 A b^3 e^3 x^3+4 a b^2 B e^3 x^3-12 b^3 B d e^2 x^3+6 a A b^2 e^3 x^2-6 a^2 b B e^3 x^2-18 A b^3 d e^2 x^2+18 a b^2 B d e^2 x^2-18 b^3 B d^2 e x^2-12 b^3 B d^3 x-12 a^2 A b e^3 x+12 a^3 B e^3 x+36 a A b^2 d e^2 x-36 a^2 b B d e^2 x-36 A b^3 d^2 e x+36 a b^2 B d^2 e x}{24 b^3 \sqrt {b^2}}+\frac {\left (-A d^3 b^5+a B d^3 b^4-A \sqrt {b^2} d^3 b^4+3 a A d^2 e b^4+a \sqrt {b^2} B d^3 b^3-3 a^2 A d e^2 b^3-3 a^2 B d^2 e b^3+3 a A \sqrt {b^2} d^2 e b^3+a^3 A e^3 b^2+3 a^3 B d e^2 b^2-a^4 B e^3 b+a^3 A \sqrt {b^2} e^3 b+3 a^3 \sqrt {b^2} B d e^2 b-a^4 \sqrt {b^2} B e^3-3 a^2 A \left (b^2\right )^{3/2} d e^2-3 a^2 \left (b^2\right )^{3/2} B d^2 e\right ) \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{2 b^5 \sqrt {b^2}}+\frac {\left (-A d^3 b^5+a B d^3 b^4+A \sqrt {b^2} d^3 b^4+3 a A d^2 e b^4-a \sqrt {b^2} B d^3 b^3-3 a^2 A d e^2 b^3-3 a^2 B d^2 e b^3-3 a A \sqrt {b^2} d^2 e b^3+a^3 A e^3 b^2+3 a^3 B d e^2 b^2-a^4 B e^3 b-a^3 A \sqrt {b^2} e^3 b-3 a^3 \sqrt {b^2} B d e^2 b+a^4 \sqrt {b^2} B e^3+3 a^2 A \left (b^2\right )^{3/2} d e^2+3 a^2 \left (b^2\right )^{3/2} B d^2 e\right ) \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{2 b^5 \sqrt {b^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 269, normalized size = 1.08 \begin {gather*} \frac {3 \, B b^{4} e^{3} x^{4} + 4 \, {\left (3 \, B b^{4} d e^{2} - {\left (B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 6 \, {\left (3 \, B b^{4} d^{2} e - 3 \, {\left (B a b^{3} - A b^{4}\right )} d e^{2} + {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 12 \, {\left (B b^{4} d^{3} - 3 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e + 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} - {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x - 12 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - {\left (B a^{4} - A a^{3} b\right )} e^{3}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 431, normalized size = 1.74 \begin {gather*} \frac {3 \, B b^{3} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, B b^{3} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 18 \, B b^{3} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + 12 \, B b^{3} d^{3} x \mathrm {sgn}\left (b x + a\right ) - 4 \, B a b^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, A b^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 18 \, B a b^{2} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 18 \, A b^{3} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 36 \, B a b^{2} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + 36 \, A b^{3} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + 6 \, B a^{2} b x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, A a b^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, B a^{2} b d x e^{2} \mathrm {sgn}\left (b x + a\right ) - 36 \, A a b^{2} d x e^{2} \mathrm {sgn}\left (b x + a\right ) - 12 \, B a^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, A a^{2} b x e^{3} \mathrm {sgn}\left (b x + a\right )}{12 \, b^{4}} - \frac {{\left (B a b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, B a^{2} b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{3} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, A a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - B a^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + A a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 356, normalized size = 1.44 \begin {gather*} -\frac {\left (b x +a \right ) \left (-3 B \,b^{4} e^{3} x^{4}-4 A \,b^{4} e^{3} x^{3}+4 B a \,b^{3} e^{3} x^{3}-12 B \,b^{4} d \,e^{2} x^{3}+6 A a \,b^{3} e^{3} x^{2}-18 A \,b^{4} d \,e^{2} x^{2}-6 B \,a^{2} b^{2} e^{3} x^{2}+18 B a \,b^{3} d \,e^{2} x^{2}-18 B \,b^{4} d^{2} e \,x^{2}+12 A \,a^{3} b \,e^{3} \ln \left (b x +a \right )-36 A \,a^{2} b^{2} d \,e^{2} \ln \left (b x +a \right )-12 A \,a^{2} b^{2} e^{3} x +36 A a \,b^{3} d^{2} e \ln \left (b x +a \right )+36 A a \,b^{3} d \,e^{2} x -12 A \,b^{4} d^{3} \ln \left (b x +a \right )-36 A \,b^{4} d^{2} e x -12 B \,a^{4} e^{3} \ln \left (b x +a \right )+36 B \,a^{3} b d \,e^{2} \ln \left (b x +a \right )+12 B \,a^{3} b \,e^{3} x -36 B \,a^{2} b^{2} d^{2} e \ln \left (b x +a \right )-36 B \,a^{2} b^{2} d \,e^{2} x +12 B a \,b^{3} d^{3} \ln \left (b x +a \right )+36 B a \,b^{3} d^{2} e x -12 B \,b^{4} d^{3} x \right )}{12 \sqrt {\left (b x +a \right )^{2}}\, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 438, normalized size = 1.77 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B e^{3} x^{3}}{4 \, b^{2}} + \frac {13 \, B a^{2} e^{3} x^{2}}{12 \, b^{3}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a e^{3} x^{2}}{12 \, b^{3}} - \frac {13 \, B a^{3} e^{3} x}{6 \, b^{4}} + \frac {A d^{3} \log \left (x + \frac {a}{b}\right )}{b} + \frac {B a^{4} e^{3} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{3} e^{3}}{6 \, b^{5}} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} a x^{2}}{6 \, b^{2}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} x^{2}}{2 \, b} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} + \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} a^{2} x}{3 \, b^{3}} - \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} a x}{b^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} a^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {2 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \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 {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 221, normalized size = 0.89 \begin {gather*} \frac {B e^{3} x^{4}}{4 b} + x^{3} \left (\frac {A e^{3}}{3 b} - \frac {B a e^{3}}{3 b^{2}} + \frac {B d e^{2}}{b}\right ) + x^{2} \left (- \frac {A a e^{3}}{2 b^{2}} + \frac {3 A d e^{2}}{2 b} + \frac {B a^{2} e^{3}}{2 b^{3}} - \frac {3 B a d e^{2}}{2 b^{2}} + \frac {3 B d^{2} e}{2 b}\right ) + x \left (\frac {A a^{2} e^{3}}{b^{3}} - \frac {3 A a d e^{2}}{b^{2}} + \frac {3 A d^{2} e}{b} - \frac {B a^{3} e^{3}}{b^{4}} + \frac {3 B a^{2} d e^{2}}{b^{3}} - \frac {3 B a d^{2} e}{b^{2}} + \frac {B d^{3}}{b}\right ) + \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{3} \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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