3.19.21 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {6 b^2 (d+e x)^{5/2} (b d-a e)}{5 e^4}+\frac {2 b (d+e x)^{3/2} (b d-a e)^2}{e^4}-\frac {2 \sqrt {d+e x} (b d-a e)^3}{e^4}+\frac {2 b^3 (d+e x)^{7/2}}{7 e^4} \]

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Rubi [A]  time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {6 b^2 (d+e x)^{5/2} (b d-a e)}{5 e^4}+\frac {2 b (d+e x)^{3/2} (b d-a e)^2}{e^4}-\frac {2 \sqrt {d+e x} (b d-a e)^3}{e^4}+\frac {2 b^3 (d+e x)^{7/2}}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2))/Sqrt[d + e*x],x]

[Out]

(-2*(b*d - a*e)^3*Sqrt[d + e*x])/e^4 + (2*b*(b*d - a*e)^2*(d + e*x)^(3/2))/e^4 - (6*b^2*(b*d - a*e)*(d + e*x)^
(5/2))/(5*e^4) + (2*b^3*(d + e*x)^(7/2))/(7*e^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^3}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^3}{e^3 \sqrt {d+e x}}+\frac {3 b (b d-a e)^2 \sqrt {d+e x}}{e^3}-\frac {3 b^2 (b d-a e) (d+e x)^{3/2}}{e^3}+\frac {b^3 (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^3 \sqrt {d+e x}}{e^4}+\frac {2 b (b d-a e)^2 (d+e x)^{3/2}}{e^4}-\frac {6 b^2 (b d-a e) (d+e x)^{5/2}}{5 e^4}+\frac {2 b^3 (d+e x)^{7/2}}{7 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 79, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-21 b^2 (d+e x)^2 (b d-a e)+35 b (d+e x) (b d-a e)^2-35 (b d-a e)^3+5 b^3 (d+e x)^3\right )}{35 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(-35*(b*d - a*e)^3 + 35*b*(b*d - a*e)^2*(d + e*x) - 21*b^2*(b*d - a*e)*(d + e*x)^2 + 5*b^3*(d
+ e*x)^3))/(35*e^4)

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IntegrateAlgebraic [A]  time = 0.06, size = 132, normalized size = 1.38 \begin {gather*} \frac {2 \sqrt {d+e x} \left (35 a^3 e^3+35 a^2 b e^2 (d+e x)-105 a^2 b d e^2+105 a b^2 d^2 e+21 a b^2 e (d+e x)^2-70 a b^2 d e (d+e x)-35 b^3 d^3+35 b^3 d^2 (d+e x)+5 b^3 (d+e x)^3-21 b^3 d (d+e x)^2\right )}{35 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(-35*b^3*d^3 + 105*a*b^2*d^2*e - 105*a^2*b*d*e^2 + 35*a^3*e^3 + 35*b^3*d^2*(d + e*x) - 70*a*b
^2*d*e*(d + e*x) + 35*a^2*b*e^2*(d + e*x) - 21*b^3*d*(d + e*x)^2 + 21*a*b^2*e*(d + e*x)^2 + 5*b^3*(d + e*x)^3)
)/(35*e^4)

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fricas [A]  time = 0.42, size = 115, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 56 \, a b^{2} d^{2} e - 70 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} - 3 \, {\left (2 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (8 \, b^{3} d^{2} e - 28 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*b^3*e^3*x^3 - 16*b^3*d^3 + 56*a*b^2*d^2*e - 70*a^2*b*d*e^2 + 35*a^3*e^3 - 3*(2*b^3*d*e^2 - 7*a*b^2*e^3
)*x^2 + (8*b^3*d^2*e - 28*a*b^2*d*e^2 + 35*a^2*b*e^3)*x)*sqrt(e*x + d)/e^4

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giac [A]  time = 0.19, size = 143, normalized size = 1.49 \begin {gather*} \frac {2}{35} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a b^{2} e^{\left (-2\right )} + {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{3} e^{\left (-3\right )} + 35 \, \sqrt {x e + d} a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/35*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*e^(-1) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15
*sqrt(x*e + d)*d^2)*a*b^2*e^(-2) + (5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqr
t(x*e + d)*d^3)*b^3*e^(-3) + 35*sqrt(x*e + d)*a^3)*e^(-1)

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maple [A]  time = 0.05, size = 116, normalized size = 1.21 \begin {gather*} \frac {2 \left (5 b^{3} e^{3} x^{3}+21 a \,b^{2} e^{3} x^{2}-6 b^{3} d \,e^{2} x^{2}+35 a^{2} b \,e^{3} x -28 a \,b^{2} d \,e^{2} x +8 b^{3} d^{2} e x +35 a^{3} e^{3}-70 a^{2} b d \,e^{2}+56 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \sqrt {e x +d}}{35 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x)

[Out]

2/35*(5*b^3*e^3*x^3+21*a*b^2*e^3*x^2-6*b^3*d*e^2*x^2+35*a^2*b*e^3*x-28*a*b^2*d*e^2*x+8*b^3*d^2*e*x+35*a^3*e^3-
70*a^2*b*d*e^2+56*a*b^2*d^2*e-16*b^3*d^3)*(e*x+d)^(1/2)/e^4

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maxima [A]  time = 0.54, size = 118, normalized size = 1.23 \begin {gather*} \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} - 21 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {e x + d}\right )}}{35 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*(e*x + d)^(7/2)*b^3 - 21*(b^3*d - a*b^2*e)*(e*x + d)^(5/2) + 35*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*(e
*x + d)^(3/2) - 35*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(e*x + d))/e^4

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mupad [B]  time = 0.06, size = 87, normalized size = 0.91 \begin {gather*} \frac {2\,b^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^4}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x))/(d + e*x)^(1/2),x)

[Out]

(2*b^3*(d + e*x)^(7/2))/(7*e^4) - ((6*b^3*d - 6*a*b^2*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(a*e - b*d)^3*(d + e*x)
^(1/2))/e^4 + (2*b*(a*e - b*d)^2*(d + e*x)^(3/2))/e^4

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sympy [A]  time = 38.31, size = 394, normalized size = 4.10 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{3} d}{\sqrt {d + e x}} - 2 a^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {6 a^{2} b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {6 a^{2} b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {6 a b^{2} d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {6 a b^{2} \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 b^{3} d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {2 b^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {a^{3} b x + \frac {3 a^{2} b^{2} x^{2}}{2} + a b^{3} x^{3} + \frac {b^{4} x^{4}}{4}}{b} & \text {otherwise} \end {cases}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*a**3*d/sqrt(d + e*x) - 2*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*a**2*b*d*(-d/sqrt(d + e*x)
- sqrt(d + e*x))/e - 6*a**2*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 6*a*b**2*d*(d
**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*a*b**2*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt
(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 2*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*
x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 2*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**
2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3)/e, Ne(e, 0)), (Piecewise((a**3*x, Eq(b
, 0)), ((a**3*b*x + 3*a**2*b**2*x**2/2 + a*b**3*x**3 + b**4*x**4/4)/b, True))/sqrt(d), True))

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