3.1969 \(\int (a+b x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=254 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11}}{11 e^5 (a+b x)}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)}{5 e^5 (a+b x)}+\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^2}{3 e^5 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^3}{2 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^4}{7 e^5 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.303381, antiderivative size = 254, 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} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11}}{11 e^5 (a+b x)}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)}{5 e^5 (a+b x)}+\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^2}{3 e^5 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^3}{2 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^4}{7 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.128092, size = 377, normalized size = 1.48 \[ \frac{x \sqrt{(a+b x)^2} \left (55 a^2 b^2 x^2 \left (756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+378 d^5 e x+84 d^6+189 d e^5 x^5+28 e^6 x^6\right )+165 a^3 b x \left (210 d^4 e^2 x^2+224 d^3 e^3 x^3+140 d^2 e^4 x^4+112 d^5 e x+28 d^6+48 d e^5 x^5+7 e^6 x^6\right )+330 a^4 \left (35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+21 d^5 e x+7 d^6+7 d e^5 x^5+e^6 x^6\right )+11 a b^3 x^3 \left (2100 d^4 e^2 x^2+2400 d^3 e^3 x^3+1575 d^2 e^4 x^4+1008 d^5 e x+210 d^6+560 d e^5 x^5+84 e^6 x^6\right )+b^4 x^4 \left (4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+2310 d^5 e x+462 d^6+1386 d e^5 x^5+210 e^6 x^6\right )\right )}{2310 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(330*a^4*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^5
*x^5 + e^6*x^6) + 165*a^3*b*x*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 + 48
*d*e^5*x^5 + 7*e^6*x^6) + 55*a^2*b^2*x^2*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e
^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 11*a*b^3*x^3*(210*d^6 + 1008*d^5*e*x + 2100*d^4*e^2*x^2 + 2400*d^3*e^3*
x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6) + b^4*x^4*(462*d^6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 57
75*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6)))/(2310*(a + b*x))

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Maple [B]  time = 0.007, size = 489, normalized size = 1.9 \begin{align*}{\frac{x \left ( 210\,{b}^{4}{e}^{6}{x}^{10}+924\,{x}^{9}a{b}^{3}{e}^{6}+1386\,{x}^{9}{b}^{4}d{e}^{5}+1540\,{x}^{8}{a}^{2}{b}^{2}{e}^{6}+6160\,{x}^{8}a{b}^{3}d{e}^{5}+3850\,{x}^{8}{b}^{4}{d}^{2}{e}^{4}+1155\,{x}^{7}{a}^{3}b{e}^{6}+10395\,{x}^{7}{a}^{2}{b}^{2}d{e}^{5}+17325\,{x}^{7}a{b}^{3}{d}^{2}{e}^{4}+5775\,{x}^{7}{b}^{4}{d}^{3}{e}^{3}+330\,{x}^{6}{a}^{4}{e}^{6}+7920\,{x}^{6}{a}^{3}bd{e}^{5}+29700\,{x}^{6}{a}^{2}{b}^{2}{d}^{2}{e}^{4}+26400\,{x}^{6}a{b}^{3}{d}^{3}{e}^{3}+4950\,{x}^{6}{b}^{4}{d}^{4}{e}^{2}+2310\,{a}^{4}d{e}^{5}{x}^{5}+23100\,{a}^{3}b{d}^{2}{e}^{4}{x}^{5}+46200\,{a}^{2}{b}^{2}{d}^{3}{e}^{3}{x}^{5}+23100\,a{b}^{3}{d}^{4}{e}^{2}{x}^{5}+2310\,{b}^{4}{d}^{5}e{x}^{5}+6930\,{x}^{4}{a}^{4}{d}^{2}{e}^{4}+36960\,{x}^{4}{a}^{3}b{d}^{3}{e}^{3}+41580\,{x}^{4}{a}^{2}{b}^{2}{d}^{4}{e}^{2}+11088\,{x}^{4}a{b}^{3}{d}^{5}e+462\,{x}^{4}{b}^{4}{d}^{6}+11550\,{a}^{4}{d}^{3}{e}^{3}{x}^{3}+34650\,{a}^{3}b{d}^{4}{e}^{2}{x}^{3}+20790\,{a}^{2}{b}^{2}{d}^{5}e{x}^{3}+2310\,a{b}^{3}{d}^{6}{x}^{3}+11550\,{a}^{4}{d}^{4}{e}^{2}{x}^{2}+18480\,{a}^{3}b{d}^{5}e{x}^{2}+4620\,{a}^{2}{b}^{2}{d}^{6}{x}^{2}+6930\,{a}^{4}{d}^{5}ex+4620\,{a}^{3}b{d}^{6}x+2310\,{a}^{4}{d}^{6} \right ) }{2310\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

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

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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)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.57153, size = 863, normalized size = 3.4 \begin{align*} \frac{1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac{1}{5} \,{\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac{1}{3} \,{\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac{1}{2} \,{\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} +{\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} +{\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} +{\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3*(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*
a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 + 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*
e^2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 + (b^4*d^5*e + 10*a*b^3*d^4*e^2 +
20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 +
80*a^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3*b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4
+ (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*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 )^{6} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**6*((a + b*x)**2)**(3/2), x)

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Giac [B]  time = 1.19039, size = 891, normalized size = 3.51 \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)^6*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

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