3.2054 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac{20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac{4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac{10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^5}{e^6}+\frac{2 b^5 (d+e x)^{11/2}}{11 e^6} \]

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x])/e^6 + (10*b*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e^6) - (4*b^2*(b*d - a*e)^3*(d
+ e*x)^(5/2))/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2))/(9*e
^6) + (2*b^5*(d + e*x)^(11/2))/(11*e^6)

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Rubi [A]  time = 0.0571058, antiderivative size = 154, normalized size of antiderivative = 1., 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 = {27, 43} \[ -\frac{10 b^4 (d+e x)^{9/2} (b d-a e)}{9 e^6}+\frac{20 b^3 (d+e x)^{7/2} (b d-a e)^2}{7 e^6}-\frac{4 b^2 (d+e x)^{5/2} (b d-a e)^3}{e^6}+\frac{10 b (d+e x)^{3/2} (b d-a e)^4}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^5}{e^6}+\frac{2 b^5 (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x])/e^6 + (10*b*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e^6) - (4*b^2*(b*d - a*e)^3*(d
+ e*x)^(5/2))/e^6 + (20*b^3*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2))/(9*e
^6) + (2*b^5*(d + e*x)^(11/2))/(11*e^6)

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 )^2}{\sqrt{d+e x}} \, dx &=\int \frac{(a+b x)^5}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{(-b d+a e)^5}{e^5 \sqrt{d+e x}}+\frac{5 b (b d-a e)^4 \sqrt{d+e x}}{e^5}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{3/2}}{e^5}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{5/2}}{e^5}-\frac{5 b^4 (b d-a e) (d+e x)^{7/2}}{e^5}+\frac{b^5 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^5 \sqrt{d+e x}}{e^6}+\frac{10 b (b d-a e)^4 (d+e x)^{3/2}}{3 e^6}-\frac{4 b^2 (b d-a e)^3 (d+e x)^{5/2}}{e^6}+\frac{20 b^3 (b d-a e)^2 (d+e x)^{7/2}}{7 e^6}-\frac{10 b^4 (b d-a e) (d+e x)^{9/2}}{9 e^6}+\frac{2 b^5 (d+e x)^{11/2}}{11 e^6}\\ \end{align*}

Mathematica [A]  time = 0.0705641, size = 123, normalized size = 0.8 \[ \frac{2 \sqrt{d+e x} \left (-1386 b^2 (d+e x)^2 (b d-a e)^3+990 b^3 (d+e x)^3 (b d-a e)^2-385 b^4 (d+e x)^4 (b d-a e)+1155 b (d+e x) (b d-a e)^4-693 (b d-a e)^5+63 b^5 (d+e x)^5\right )}{693 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-693*(b*d - a*e)^5 + 1155*b*(b*d - a*e)^4*(d + e*x) - 1386*b^2*(b*d - a*e)^3*(d + e*x)^2 + 9
90*b^3*(b*d - a*e)^2*(d + e*x)^3 - 385*b^4*(b*d - a*e)*(d + e*x)^4 + 63*b^5*(d + e*x)^5))/(693*e^6)

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Maple [B]  time = 0.004, size = 273, normalized size = 1.8 \begin{align*}{\frac{126\,{x}^{5}{b}^{5}{e}^{5}+770\,{x}^{4}a{b}^{4}{e}^{5}-140\,{x}^{4}{b}^{5}d{e}^{4}+1980\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-880\,{x}^{3}a{b}^{4}d{e}^{4}+160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+2772\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-2376\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+1056\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+2310\,x{a}^{4}b{e}^{5}-3696\,x{a}^{3}{b}^{2}d{e}^{4}+3168\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-1408\,xa{b}^{4}{d}^{3}{e}^{2}+256\,x{b}^{5}{d}^{4}e+1386\,{a}^{5}{e}^{5}-4620\,{a}^{4}bd{e}^{4}+7392\,{a}^{3}{d}^{2}{b}^{2}{e}^{3}-6336\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2816\,a{d}^{4}{b}^{4}e-512\,{b}^{5}{d}^{5}}{693\,{e}^{6}}\sqrt{ex+d}} \end{align*}

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

[Out]

2/693*(63*b^5*e^5*x^5+385*a*b^4*e^5*x^4-70*b^5*d*e^4*x^4+990*a^2*b^3*e^5*x^3-440*a*b^4*d*e^4*x^3+80*b^5*d^2*e^
3*x^3+1386*a^3*b^2*e^5*x^2-1188*a^2*b^3*d*e^4*x^2+528*a*b^4*d^2*e^3*x^2-96*b^5*d^3*e^2*x^2+1155*a^4*b*e^5*x-18
48*a^3*b^2*d*e^4*x+1584*a^2*b^3*d^2*e^3*x-704*a*b^4*d^3*e^2*x+128*b^5*d^4*e*x+693*a^5*e^5-2310*a^4*b*d*e^4+369
6*a^3*b^2*d^2*e^3-3168*a^2*b^3*d^3*e^2+1408*a*b^4*d^4*e-256*b^5*d^5)*(e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.969323, size = 350, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{5} - 385 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 990 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 693 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{e x + d}\right )}}{693 \, e^{6}} \end{align*}

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

[Out]

2/693*(63*(e*x + d)^(11/2)*b^5 - 385*(b^5*d - a*b^4*e)*(e*x + d)^(9/2) + 990*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*
e^2)*(e*x + d)^(7/2) - 1386*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(5/2) + 1155*(
b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(3/2) - 693*(b^5*d^5 - 5*
a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(e*x + d))/e^6

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Fricas [A]  time = 1.39303, size = 586, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{6}} \end{align*}

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

[Out]

2/693*(63*b^5*e^5*x^5 - 256*b^5*d^5 + 1408*a*b^4*d^4*e - 3168*a^2*b^3*d^3*e^2 + 3696*a^3*b^2*d^2*e^3 - 2310*a^
4*b*d*e^4 + 693*a^5*e^5 - 35*(2*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 44*a*b^4*d*e^4 + 99*a^2*b^
3*e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 88*a*b^4*d^2*e^3 + 198*a^2*b^3*d*e^4 - 231*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e
- 704*a*b^4*d^3*e^2 + 1584*a^2*b^3*d^2*e^3 - 1848*a^3*b^2*d*e^4 + 1155*a^4*b*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 61.476, size = 740, normalized size = 4.81 \begin{align*} \text{result too large to display} \end{align*}

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)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a**5*d/sqrt(d + e*x) + 2*a**5*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 10*a**4*b*d*(-d/sqrt(d + e*x
) - sqrt(d + e*x))/e + 10*a**4*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 20*a**3*b**
2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 20*a**3*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 + 20*a**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 + 20*a**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 + 10*a*b**4*d*(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**4 + 10*a*b**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x
)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 2*b**5*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d +
e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e*
*5 + 2*b**5*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 1
5*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), (Piecewise((a**
5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True))/sqrt(d), True))

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Giac [B]  time = 1.13341, size = 402, normalized size = 2.61 \begin{align*} \frac{2}{693} \,{\left (1155 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{4} b e^{\left (-1\right )} + 462 \,{\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^{3} b^{2} e^{\left (-2\right )} + 198 \,{\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 )} a^{2} b^{3} e^{\left (-3\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} a b^{4} e^{\left (-4\right )} +{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} b^{5} e^{\left (-5\right )} + 693 \, \sqrt{x e + d} a^{5}\right )} e^{\left (-1\right )} \end{align*}

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

[Out]

2/693*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*e^(-1) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*a^3*b^2*e^(-2) + 198*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d
^2 - 35*sqrt(x*e + d)*d^3)*a^2*b^3*e^(-3) + 11*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/
2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^4*e^(-4) + (63*(x*e + d)^(11/2) - 385*(x*e + d)^
(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^
5)*b^5*e^(-5) + 693*sqrt(x*e + d)*a^5)*e^(-1)