3.2061 \(\int (a+b x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=214 \[ -\frac{14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac{42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac{70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac{70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac{6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac{14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac{2 b^7 (d+e x)^{17/2}}{17 e^8} \]

[Out]

(-2*(b*d - a*e)^7*(d + e*x)^(3/2))/(3*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^8) - (6*b^2*(b*d - a*e)
^5*(d + e*x)^(7/2))/e^8 + (70*b^3*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(11
/2))/(11*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(15/2))/(15*e
^8) + (2*b^7*(d + e*x)^(17/2))/(17*e^8)

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Rubi [A]  time = 0.0727237, antiderivative size = 214, 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} \[ -\frac{14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac{42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac{70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac{70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac{6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac{14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac{2 b^7 (d+e x)^{17/2}}{17 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^7*(d + e*x)^(3/2))/(3*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^8) - (6*b^2*(b*d - a*e)
^5*(d + e*x)^(7/2))/e^8 + (70*b^3*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(11
/2))/(11*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(15/2))/(15*e
^8) + (2*b^7*(d + e*x)^(17/2))/(17*e^8)

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 (a+b x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{(-b d+a e)^7 \sqrt{d+e x}}{e^7}+\frac{7 b (b d-a e)^6 (d+e x)^{3/2}}{e^7}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{5/2}}{e^7}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{9/2}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{11/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{13/2}}{e^7}+\frac{b^7 (d+e x)^{15/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^7 (d+e x)^{3/2}}{3 e^8}+\frac{14 b (b d-a e)^6 (d+e x)^{5/2}}{5 e^8}-\frac{6 b^2 (b d-a e)^5 (d+e x)^{7/2}}{e^8}+\frac{70 b^3 (b d-a e)^4 (d+e x)^{9/2}}{9 e^8}-\frac{70 b^4 (b d-a e)^3 (d+e x)^{11/2}}{11 e^8}+\frac{42 b^5 (b d-a e)^2 (d+e x)^{13/2}}{13 e^8}-\frac{14 b^6 (b d-a e) (d+e x)^{15/2}}{15 e^8}+\frac{2 b^7 (d+e x)^{17/2}}{17 e^8}\\ \end{align*}

Mathematica [A]  time = 0.129168, size = 167, normalized size = 0.78 \[ \frac{2 (d+e x)^{3/2} \left (-328185 b^2 (d+e x)^2 (b d-a e)^5+425425 b^3 (d+e x)^3 (b d-a e)^4-348075 b^4 (d+e x)^4 (b d-a e)^3+176715 b^5 (d+e x)^5 (b d-a e)^2-51051 b^6 (d+e x)^6 (b d-a e)+153153 b (d+e x) (b d-a e)^6-36465 (b d-a e)^7+6435 b^7 (d+e x)^7\right )}{109395 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-36465*(b*d - a*e)^7 + 153153*b*(b*d - a*e)^6*(d + e*x) - 328185*b^2*(b*d - a*e)^5*(d + e*
x)^2 + 425425*b^3*(b*d - a*e)^4*(d + e*x)^3 - 348075*b^4*(b*d - a*e)^3*(d + e*x)^4 + 176715*b^5*(b*d - a*e)^2*
(d + e*x)^5 - 51051*b^6*(b*d - a*e)*(d + e*x)^6 + 6435*b^7*(d + e*x)^7))/(109395*e^8)

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Maple [B]  time = 0.005, size = 498, normalized size = 2.3 \begin{align*}{\frac{12870\,{b}^{7}{x}^{7}{e}^{7}+102102\,a{b}^{6}{e}^{7}{x}^{6}-12012\,{b}^{7}d{e}^{6}{x}^{6}+353430\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}-94248\,a{b}^{6}d{e}^{6}{x}^{5}+11088\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}+696150\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}-321300\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}+85680\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}-10080\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}+850850\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}-618800\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}+285600\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}-76160\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}+8960\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+656370\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-729300\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+530400\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-244800\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+65280\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-7680\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+306306\,{a}^{6}b{e}^{7}x-525096\,{a}^{5}{b}^{2}d{e}^{6}x+583440\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-424320\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+195840\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-52224\,a{b}^{6}{d}^{5}{e}^{2}x+6144\,{b}^{7}{d}^{6}ex+72930\,{a}^{7}{e}^{7}-204204\,{a}^{6}bd{e}^{6}+350064\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-388960\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+282880\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-130560\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+34816\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{109395\,{e}^{8}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \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)^3*(e*x+d)^(1/2),x)

[Out]

2/109395*(e*x+d)^(3/2)*(6435*b^7*e^7*x^7+51051*a*b^6*e^7*x^6-6006*b^7*d*e^6*x^6+176715*a^2*b^5*e^7*x^5-47124*a
*b^6*d*e^6*x^5+5544*b^7*d^2*e^5*x^5+348075*a^3*b^4*e^7*x^4-160650*a^2*b^5*d*e^6*x^4+42840*a*b^6*d^2*e^5*x^4-50
40*b^7*d^3*e^4*x^4+425425*a^4*b^3*e^7*x^3-309400*a^3*b^4*d*e^6*x^3+142800*a^2*b^5*d^2*e^5*x^3-38080*a*b^6*d^3*
e^4*x^3+4480*b^7*d^4*e^3*x^3+328185*a^5*b^2*e^7*x^2-364650*a^4*b^3*d*e^6*x^2+265200*a^3*b^4*d^2*e^5*x^2-122400
*a^2*b^5*d^3*e^4*x^2+32640*a*b^6*d^4*e^3*x^2-3840*b^7*d^5*e^2*x^2+153153*a^6*b*e^7*x-262548*a^5*b^2*d*e^6*x+29
1720*a^4*b^3*d^2*e^5*x-212160*a^3*b^4*d^3*e^4*x+97920*a^2*b^5*d^4*e^3*x-26112*a*b^6*d^5*e^2*x+3072*b^7*d^6*e*x
+36465*a^7*e^7-102102*a^6*b*d*e^6+175032*a^5*b^2*d^2*e^5-194480*a^4*b^3*d^3*e^4+141440*a^3*b^4*d^4*e^3-65280*a
^2*b^5*d^5*e^2+17408*a*b^6*d^6*e-2048*b^7*d^7)/e^8

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Maxima [B]  time = 0.990488, size = 616, normalized size = 2.88 \begin{align*} \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{7} - 51051 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 176715 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 348075 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 425425 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 328185 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 153153 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 36465 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{109395 \, e^{8}} \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)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*b^7 - 51051*(b^7*d - a*b^6*e)*(e*x + d)^(15/2) + 176715*(b^7*d^2 - 2*a*b^6*d*e
+ a^2*b^5*e^2)*(e*x + d)^(13/2) - 348075*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d)^
(11/2) + 425425*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*(e*x + d)^(9/2)
- 328185*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*(
e*x + d)^(7/2) + 153153*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^
4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d)^(5/2) - 36465*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^
3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d)^(3/2))/e^8

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Fricas [B]  time = 1.32321, size = 1323, normalized size = 6.18 \begin{align*} \frac{2 \,{\left (6435 \, b^{7} e^{8} x^{8} - 2048 \, b^{7} d^{8} + 17408 \, a b^{6} d^{7} e - 65280 \, a^{2} b^{5} d^{6} e^{2} + 141440 \, a^{3} b^{4} d^{5} e^{3} - 194480 \, a^{4} b^{3} d^{4} e^{4} + 175032 \, a^{5} b^{2} d^{3} e^{5} - 102102 \, a^{6} b d^{2} e^{6} + 36465 \, a^{7} d e^{7} + 429 \,{\left (b^{7} d e^{7} + 119 \, a b^{6} e^{8}\right )} x^{7} - 231 \,{\left (2 \, b^{7} d^{2} e^{6} - 17 \, a b^{6} d e^{7} - 765 \, a^{2} b^{5} e^{8}\right )} x^{6} + 63 \,{\left (8 \, b^{7} d^{3} e^{5} - 68 \, a b^{6} d^{2} e^{6} + 255 \, a^{2} b^{5} d e^{7} + 5525 \, a^{3} b^{4} e^{8}\right )} x^{5} - 35 \,{\left (16 \, b^{7} d^{4} e^{4} - 136 \, a b^{6} d^{3} e^{5} + 510 \, a^{2} b^{5} d^{2} e^{6} - 1105 \, a^{3} b^{4} d e^{7} - 12155 \, a^{4} b^{3} e^{8}\right )} x^{4} + 5 \,{\left (128 \, b^{7} d^{5} e^{3} - 1088 \, a b^{6} d^{4} e^{4} + 4080 \, a^{2} b^{5} d^{3} e^{5} - 8840 \, a^{3} b^{4} d^{2} e^{6} + 12155 \, a^{4} b^{3} d e^{7} + 65637 \, a^{5} b^{2} e^{8}\right )} x^{3} - 3 \,{\left (256 \, b^{7} d^{6} e^{2} - 2176 \, a b^{6} d^{5} e^{3} + 8160 \, a^{2} b^{5} d^{4} e^{4} - 17680 \, a^{3} b^{4} d^{3} e^{5} + 24310 \, a^{4} b^{3} d^{2} e^{6} - 21879 \, a^{5} b^{2} d e^{7} - 51051 \, a^{6} b e^{8}\right )} x^{2} +{\left (1024 \, b^{7} d^{7} e - 8704 \, a b^{6} d^{6} e^{2} + 32640 \, a^{2} b^{5} d^{5} e^{3} - 70720 \, a^{3} b^{4} d^{4} e^{4} + 97240 \, a^{4} b^{3} d^{3} e^{5} - 87516 \, a^{5} b^{2} d^{2} e^{6} + 51051 \, a^{6} b d e^{7} + 36465 \, a^{7} e^{8}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{8}} \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)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/109395*(6435*b^7*e^8*x^8 - 2048*b^7*d^8 + 17408*a*b^6*d^7*e - 65280*a^2*b^5*d^6*e^2 + 141440*a^3*b^4*d^5*e^3
- 194480*a^4*b^3*d^4*e^4 + 175032*a^5*b^2*d^3*e^5 - 102102*a^6*b*d^2*e^6 + 36465*a^7*d*e^7 + 429*(b^7*d*e^7 +
119*a*b^6*e^8)*x^7 - 231*(2*b^7*d^2*e^6 - 17*a*b^6*d*e^7 - 765*a^2*b^5*e^8)*x^6 + 63*(8*b^7*d^3*e^5 - 68*a*b^
6*d^2*e^6 + 255*a^2*b^5*d*e^7 + 5525*a^3*b^4*e^8)*x^5 - 35*(16*b^7*d^4*e^4 - 136*a*b^6*d^3*e^5 + 510*a^2*b^5*d
^2*e^6 - 1105*a^3*b^4*d*e^7 - 12155*a^4*b^3*e^8)*x^4 + 5*(128*b^7*d^5*e^3 - 1088*a*b^6*d^4*e^4 + 4080*a^2*b^5*
d^3*e^5 - 8840*a^3*b^4*d^2*e^6 + 12155*a^4*b^3*d*e^7 + 65637*a^5*b^2*e^8)*x^3 - 3*(256*b^7*d^6*e^2 - 2176*a*b^
6*d^5*e^3 + 8160*a^2*b^5*d^4*e^4 - 17680*a^3*b^4*d^3*e^5 + 24310*a^4*b^3*d^2*e^6 - 21879*a^5*b^2*d*e^7 - 51051
*a^6*b*e^8)*x^2 + (1024*b^7*d^7*e - 8704*a*b^6*d^6*e^2 + 32640*a^2*b^5*d^5*e^3 - 70720*a^3*b^4*d^4*e^4 + 97240
*a^4*b^3*d^3*e^5 - 87516*a^5*b^2*d^2*e^6 + 51051*a^6*b*d*e^7 + 36465*a^7*e^8)*x)*sqrt(e*x + d)/e^8

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Sympy [B]  time = 7.97754, size = 544, normalized size = 2.54 \begin{align*} \frac{2 \left (\frac{b^{7} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{7}} + \frac{\left (d + e x\right )^{\frac{15}{2}} \left (7 a b^{6} e - 7 b^{7} d\right )}{15 e^{7}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{13 e^{7}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{11 e^{7}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{9 e^{7}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{3 e^{7}}\right )}{e} \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)**3*(e*x+d)**(1/2),x)

[Out]

2*(b**7*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(7*a*b**6*e - 7*b**7*d)/(15*e**7) + (d + e*x)**(13/2)*
(21*a**2*b**5*e**2 - 42*a*b**6*d*e + 21*b**7*d**2)/(13*e**7) + (d + e*x)**(11/2)*(35*a**3*b**4*e**3 - 105*a**2
*b**5*d*e**2 + 105*a*b**6*d**2*e - 35*b**7*d**3)/(11*e**7) + (d + e*x)**(9/2)*(35*a**4*b**3*e**4 - 140*a**3*b*
*4*d*e**3 + 210*a**2*b**5*d**2*e**2 - 140*a*b**6*d**3*e + 35*b**7*d**4)/(9*e**7) + (d + e*x)**(7/2)*(21*a**5*b
**2*e**5 - 105*a**4*b**3*d*e**4 + 210*a**3*b**4*d**2*e**3 - 210*a**2*b**5*d**3*e**2 + 105*a*b**6*d**4*e - 21*b
**7*d**5)/(7*e**7) + (d + e*x)**(5/2)*(7*a**6*b*e**6 - 42*a**5*b**2*d*e**5 + 105*a**4*b**3*d**2*e**4 - 140*a**
3*b**4*d**3*e**3 + 105*a**2*b**5*d**4*e**2 - 42*a*b**6*d**5*e + 7*b**7*d**6)/(5*e**7) + (d + e*x)**(3/2)*(a**7
*e**7 - 7*a**6*b*d*e**6 + 21*a**5*b**2*d**2*e**5 - 35*a**4*b**3*d**3*e**4 + 35*a**3*b**4*d**4*e**3 - 21*a**2*b
**5*d**5*e**2 + 7*a*b**6*d**6*e - b**7*d**7)/(3*e**7))/e

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Giac [B]  time = 1.14808, size = 684, normalized size = 3.2 \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)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/109395*(51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*b*e^(-1) + 21879*(15*(x*e + d)^(7/2) - 42*(x*e +
d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b^2*e^(-2) + 12155*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189
*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^4*b^3*e^(-3) + 1105*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(
9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^3*b^4*e^(-4) + 255*
(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009
*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^2*b^5*e^(-5) + 17*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^
(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e +
d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a*b^6*e^(-6) + (6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 1
76715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)
*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*b^7*e^(-7) + 36465*(x*e + d)^(3/2)*a^7)*e^(-1)