3.1995 \(\int (a+b x) (d+e x)^2 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=125 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{4 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.179714, antiderivative size = 125, 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} \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{4 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0808995, size = 217, normalized size = 1.74 \[ \frac{x \sqrt{(a+b x)^2} \left (126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )}{252 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 126*a^4
*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 84*a^3*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(2
1*d^2 + 35*d*e*x + 15*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + b^6*x^6*(36*d^2 + 63*d*e*x + 2
8*e^2*x^2)))/(252*(a + b*x))

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Maple [B]  time = 0.006, size = 271, normalized size = 2.2 \begin{align*}{\frac{x \left ( 28\,{e}^{2}{b}^{6}{x}^{8}+189\,{x}^{7}{e}^{2}a{b}^{5}+63\,{x}^{7}de{b}^{6}+540\,{x}^{6}{e}^{2}{a}^{2}{b}^{4}+432\,{x}^{6}dea{b}^{5}+36\,{x}^{6}{d}^{2}{b}^{6}+840\,{x}^{5}{e}^{2}{a}^{3}{b}^{3}+1260\,{x}^{5}de{a}^{2}{b}^{4}+252\,{x}^{5}{d}^{2}a{b}^{5}+756\,{a}^{4}{b}^{2}{e}^{2}{x}^{4}+2016\,{a}^{3}{b}^{3}de{x}^{4}+756\,{a}^{2}{b}^{4}{d}^{2}{x}^{4}+378\,{x}^{3}{e}^{2}{a}^{5}b+1890\,{x}^{3}de{a}^{4}{b}^{2}+1260\,{x}^{3}{d}^{2}{a}^{3}{b}^{3}+84\,{x}^{2}{e}^{2}{a}^{6}+1008\,{x}^{2}de{a}^{5}b+1260\,{x}^{2}{d}^{2}{a}^{4}{b}^{2}+252\,{a}^{6}dex+756\,{a}^{5}b{d}^{2}x+252\,{d}^{2}{a}^{6} \right ) }{252\, \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)^2*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/252*x*(28*b^6*e^2*x^8+189*a*b^5*e^2*x^7+63*b^6*d*e*x^7+540*a^2*b^4*e^2*x^6+432*a*b^5*d*e*x^6+36*b^6*d^2*x^6+
840*a^3*b^3*e^2*x^5+1260*a^2*b^4*d*e*x^5+252*a*b^5*d^2*x^5+756*a^4*b^2*e^2*x^4+2016*a^3*b^3*d*e*x^4+756*a^2*b^
4*d^2*x^4+378*a^5*b*e^2*x^3+1890*a^4*b^2*d*e*x^3+1260*a^3*b^3*d^2*x^3+84*a^6*e^2*x^2+1008*a^5*b*d*e*x^2+1260*a
^4*b^2*d^2*x^2+252*a^6*d*e*x+756*a^5*b*d^2*x+252*a^6*d^2)*((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)^2*(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 = 1.51087, size = 495, normalized size = 3.96 \begin{align*} \frac{1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac{1}{4} \,{\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^6*e^2*x^9 + a^6*d^2*x + 1/4*(b^6*d*e + 3*a*b^5*e^2)*x^8 + 1/7*(b^6*d^2 + 12*a*b^5*d*e + 15*a^2*b^4*e^2)*
x^7 + 1/3*(3*a*b^5*d^2 + 15*a^2*b^4*d*e + 10*a^3*b^3*e^2)*x^6 + (3*a^2*b^4*d^2 + 8*a^3*b^3*d*e + 3*a^4*b^2*e^2
)*x^5 + 1/2*(10*a^3*b^3*d^2 + 15*a^4*b^2*d*e + 3*a^5*b*e^2)*x^4 + 1/3*(15*a^4*b^2*d^2 + 12*a^5*b*d*e + a^6*e^2
)*x^3 + (3*a^5*b*d^2 + a^6*d*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 )^{2} \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)**2*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.13854, size = 512, normalized size = 4.1 \begin{align*} \frac{1}{9} \, b^{6} x^{9} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, b^{6} d x^{8} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, b^{6} d^{2} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, a b^{5} x^{8} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{12}{7} \, a b^{5} d x^{7} e \mathrm{sgn}\left (b x + a\right ) + a b^{5} d^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{7} \, a^{2} b^{4} x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{2} b^{4} d x^{6} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{3} x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, a^{3} b^{3} d x^{5} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{4} b^{2} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{2} \, a^{4} b^{2} d x^{4} e \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{5} b x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{5} b d x^{3} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, a^{6} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + a^{6} d x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{6} d^{2} x \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^6*x^9*e^2*sgn(b*x + a) + 1/4*b^6*d*x^8*e*sgn(b*x + a) + 1/7*b^6*d^2*x^7*sgn(b*x + a) + 3/4*a*b^5*x^8*e^2
*sgn(b*x + a) + 12/7*a*b^5*d*x^7*e*sgn(b*x + a) + a*b^5*d^2*x^6*sgn(b*x + a) + 15/7*a^2*b^4*x^7*e^2*sgn(b*x +
a) + 5*a^2*b^4*d*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^2*x^5*sgn(b*x + a) + 10/3*a^3*b^3*x^6*e^2*sgn(b*x + a) + 8*a
^3*b^3*d*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^2*x^4*sgn(b*x + a) + 3*a^4*b^2*x^5*e^2*sgn(b*x + a) + 15/2*a^4*b^2*d
*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^2*x^3*sgn(b*x + a) + 3/2*a^5*b*x^4*e^2*sgn(b*x + a) + 4*a^5*b*d*x^3*e*sgn(b*
x + a) + 3*a^5*b*d^2*x^2*sgn(b*x + a) + 1/3*a^6*x^3*e^2*sgn(b*x + a) + a^6*d*x^2*e*sgn(b*x + a) + a^6*d^2*x*sg
n(b*x + a)