3.1993 \(\int (a+b x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=219 \[ \frac{2 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{5 b^5}+\frac{2 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{3 b^5}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{2 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^5} \]

[Out]

((b*d - a*e)^4*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^5) + (e*(b*d - a*e)^3*(a + b*x)^7*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(2*b^5) + (2*e^2*(b*d - a*e)^2*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^5) + (2*e^3*(
b*d - a*e)*(a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*b^5) + (e^4*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(11*b^5)

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Rubi [A]  time = 0.304315, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{2 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{5 b^5}+\frac{2 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{3 b^5}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{2 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^4*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^5) + (e*(b*d - a*e)^3*(a + b*x)^7*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(2*b^5) + (2*e^2*(b*d - a*e)^2*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^5) + (2*e^3*(
b*d - a*e)*(a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*b^5) + (e^4*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(11*b^5)

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^4 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^4 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(b d-a e)^4 (a+b x)^6}{b^4}+\frac{4 e (b d-a e)^3 (a+b x)^7}{b^4}+\frac{6 e^2 (b d-a e)^2 (a+b x)^8}{b^4}+\frac{4 e^3 (b d-a e) (a+b x)^9}{b^4}+\frac{e^4 (a+b x)^{10}}{b^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^4 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac{e (b d-a e)^3 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac{2 e^2 (b d-a e)^2 (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{3 b^5}+\frac{2 e^3 (b d-a e) (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac{e^4 (a+b x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{11 b^5}\\ \end{align*}

Mathematica [A]  time = 0.128374, size = 371, normalized size = 1.69 \[ \frac{x \sqrt{(a+b x)^2} \left (330 a^4 b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+165 a^3 b^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+55 a^2 b^4 x^4 \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+462 a^5 b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+462 a^6 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+11 a b^5 x^5 \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )+b^6 x^6 \left (1540 d^2 e^2 x^2+1155 d^3 e x+330 d^4+924 d e^3 x^3+210 e^4 x^4\right )\right )}{2310 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(462*a^6*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 462*a^5*b*x*(15*
d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 330*a^4*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^
2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 165*a^3*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x
^3 + 35*e^4*x^4) + 55*a^2*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + 11*
a*b^5*x^5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + b^6*x^6*(330*d^4 + 1155*d^
3*e*x + 1540*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4)))/(2310*(a + b*x))

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Maple [B]  time = 0.007, size = 489, normalized size = 2.2 \begin{align*}{\frac{x \left ( 210\,{e}^{4}{b}^{6}{x}^{10}+1386\,{x}^{9}{e}^{4}{b}^{5}a+924\,{x}^{9}d{e}^{3}{b}^{6}+3850\,{x}^{8}{e}^{4}{a}^{2}{b}^{4}+6160\,{x}^{8}d{e}^{3}{b}^{5}a+1540\,{x}^{8}{d}^{2}{e}^{2}{b}^{6}+5775\,{x}^{7}{e}^{4}{a}^{3}{b}^{3}+17325\,{x}^{7}d{e}^{3}{a}^{2}{b}^{4}+10395\,{x}^{7}{d}^{2}{e}^{2}{b}^{5}a+1155\,{x}^{7}{d}^{3}e{b}^{6}+4950\,{x}^{6}{e}^{4}{a}^{4}{b}^{2}+26400\,{x}^{6}d{e}^{3}{a}^{3}{b}^{3}+29700\,{x}^{6}{d}^{2}{e}^{2}{a}^{2}{b}^{4}+7920\,{x}^{6}{d}^{3}e{b}^{5}a+330\,{x}^{6}{d}^{4}{b}^{6}+2310\,{a}^{5}b{e}^{4}{x}^{5}+23100\,{a}^{4}{b}^{2}d{e}^{3}{x}^{5}+46200\,{a}^{3}{b}^{3}{d}^{2}{e}^{2}{x}^{5}+23100\,{a}^{2}{b}^{4}{d}^{3}e{x}^{5}+2310\,a{b}^{5}{d}^{4}{x}^{5}+462\,{x}^{4}{e}^{4}{a}^{6}+11088\,{x}^{4}d{e}^{3}{a}^{5}b+41580\,{x}^{4}{d}^{2}{e}^{2}{a}^{4}{b}^{2}+36960\,{x}^{4}{d}^{3}e{a}^{3}{b}^{3}+6930\,{x}^{4}{d}^{4}{a}^{2}{b}^{4}+2310\,{a}^{6}d{e}^{3}{x}^{3}+20790\,{a}^{5}b{d}^{2}{e}^{2}{x}^{3}+34650\,{a}^{4}{b}^{2}{d}^{3}e{x}^{3}+11550\,{a}^{3}{b}^{3}{d}^{4}{x}^{3}+4620\,{a}^{6}{d}^{2}{e}^{2}{x}^{2}+18480\,{a}^{5}b{d}^{3}e{x}^{2}+11550\,{a}^{4}{b}^{2}{d}^{4}{x}^{2}+4620\,{a}^{6}{d}^{3}ex+6930\,{a}^{5}b{d}^{4}x+2310\,{d}^{4}{a}^{6} \right ) }{2310\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/2310*x*(210*b^6*e^4*x^10+1386*a*b^5*e^4*x^9+924*b^6*d*e^3*x^9+3850*a^2*b^4*e^4*x^8+6160*a*b^5*d*e^3*x^8+1540
*b^6*d^2*e^2*x^8+5775*a^3*b^3*e^4*x^7+17325*a^2*b^4*d*e^3*x^7+10395*a*b^5*d^2*e^2*x^7+1155*b^6*d^3*e*x^7+4950*
a^4*b^2*e^4*x^6+26400*a^3*b^3*d*e^3*x^6+29700*a^2*b^4*d^2*e^2*x^6+7920*a*b^5*d^3*e*x^6+330*b^6*d^4*x^6+2310*a^
5*b*e^4*x^5+23100*a^4*b^2*d*e^3*x^5+46200*a^3*b^3*d^2*e^2*x^5+23100*a^2*b^4*d^3*e*x^5+2310*a*b^5*d^4*x^5+462*a
^6*e^4*x^4+11088*a^5*b*d*e^3*x^4+41580*a^4*b^2*d^2*e^2*x^4+36960*a^3*b^3*d^3*e*x^4+6930*a^2*b^4*d^4*x^4+2310*a
^6*d*e^3*x^3+20790*a^5*b*d^2*e^2*x^3+34650*a^4*b^2*d^3*e*x^3+11550*a^3*b^3*d^4*x^3+4620*a^6*d^2*e^2*x^2+18480*
a^5*b*d^3*e*x^2+11550*a^4*b^2*d^4*x^2+4620*a^6*d^3*e*x+6930*a^5*b*d^4*x+2310*a^6*d^4)*((b*x+a)^2)^(5/2)/(b*x+a
)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.49713, size = 863, normalized size = 3.94 \begin{align*} \frac{1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac{1}{5} \,{\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac{1}{2} \,{\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} +{\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} +{\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/11*b^6*e^4*x^11 + a^6*d^4*x + 1/5*(2*b^6*d*e^3 + 3*a*b^5*e^4)*x^10 + 1/3*(2*b^6*d^2*e^2 + 8*a*b^5*d*e^3 + 5*
a^2*b^4*e^4)*x^9 + 1/2*(b^6*d^3*e + 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 + 5*a^3*b^3*e^4)*x^8 + 1/7*(b^6*d^4 + 2
4*a*b^5*d^3*e + 90*a^2*b^4*d^2*e^2 + 80*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x^7 + (a*b^5*d^4 + 10*a^2*b^4*d^3*e +
20*a^3*b^3*d^2*e^2 + 10*a^4*b^2*d*e^3 + a^5*b*e^4)*x^6 + 1/5*(15*a^2*b^4*d^4 + 80*a^3*b^3*d^3*e + 90*a^4*b^2*d
^2*e^2 + 24*a^5*b*d*e^3 + a^6*e^4)*x^5 + (5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2 + a^6*d*e^3)*x^4
+ (5*a^4*b^2*d^4 + 8*a^5*b*d^3*e + 2*a^6*d^2*e^2)*x^3 + (3*a^5*b*d^4 + 2*a^6*d^3*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right ) \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**4*((a + b*x)**2)**(5/2), x)

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Giac [B]  time = 1.18334, size = 899, normalized size = 4.11 \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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/11*b^6*x^11*e^4*sgn(b*x + a) + 2/5*b^6*d*x^10*e^3*sgn(b*x + a) + 2/3*b^6*d^2*x^9*e^2*sgn(b*x + a) + 1/2*b^6*
d^3*x^8*e*sgn(b*x + a) + 1/7*b^6*d^4*x^7*sgn(b*x + a) + 3/5*a*b^5*x^10*e^4*sgn(b*x + a) + 8/3*a*b^5*d*x^9*e^3*
sgn(b*x + a) + 9/2*a*b^5*d^2*x^8*e^2*sgn(b*x + a) + 24/7*a*b^5*d^3*x^7*e*sgn(b*x + a) + a*b^5*d^4*x^6*sgn(b*x
+ a) + 5/3*a^2*b^4*x^9*e^4*sgn(b*x + a) + 15/2*a^2*b^4*d*x^8*e^3*sgn(b*x + a) + 90/7*a^2*b^4*d^2*x^7*e^2*sgn(b
*x + a) + 10*a^2*b^4*d^3*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^4*x^5*sgn(b*x + a) + 5/2*a^3*b^3*x^8*e^4*sgn(b*x + a
) + 80/7*a^3*b^3*d*x^7*e^3*sgn(b*x + a) + 20*a^3*b^3*d^2*x^6*e^2*sgn(b*x + a) + 16*a^3*b^3*d^3*x^5*e*sgn(b*x +
a) + 5*a^3*b^3*d^4*x^4*sgn(b*x + a) + 15/7*a^4*b^2*x^7*e^4*sgn(b*x + a) + 10*a^4*b^2*d*x^6*e^3*sgn(b*x + a) +
18*a^4*b^2*d^2*x^5*e^2*sgn(b*x + a) + 15*a^4*b^2*d^3*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^4*x^3*sgn(b*x + a) + a^
5*b*x^6*e^4*sgn(b*x + a) + 24/5*a^5*b*d*x^5*e^3*sgn(b*x + a) + 9*a^5*b*d^2*x^4*e^2*sgn(b*x + a) + 8*a^5*b*d^3*
x^3*e*sgn(b*x + a) + 3*a^5*b*d^4*x^2*sgn(b*x + a) + 1/5*a^6*x^5*e^4*sgn(b*x + a) + a^6*d*x^4*e^3*sgn(b*x + a)
+ 2*a^6*d^2*x^3*e^2*sgn(b*x + a) + 2*a^6*d^3*x^2*e*sgn(b*x + a) + a^6*d^4*x*sgn(b*x + a)