3.1441 \(\int \frac{(A+B x) (a+c x^2)^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=348 \[ -\frac{2 c (d+e x)^{7/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac{6 c^2 (d+e x)^{11/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac{2 c^2 (d+e x)^{9/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}-\frac{6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8}-\frac{2 c^3 (d+e x)^{13/2} (7 B d-A e)}{13 e^8}+\frac{2 B c^3 (d+e x)^{15/2}}{15 e^8} \]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^8 + (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*
(d + e*x)^(3/2))/(3*e^8) - (6*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(5
/2))/(5*e^8) - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(7
/2))/(7*e^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(9/2))/(9*e^8) + (6*c^2
*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(11/2))/(11*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(13/2))/(13*e^8
) + (2*B*c^3*(d + e*x)^(15/2))/(15*e^8)

________________________________________________________________________________________

Rubi [A]  time = 0.205371, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {772} \[ -\frac{2 c (d+e x)^{7/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac{6 c^2 (d+e x)^{11/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac{2 c^2 (d+e x)^{9/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}-\frac{6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8}-\frac{2 c^3 (d+e x)^{13/2} (7 B d-A e)}{13 e^8}+\frac{2 B c^3 (d+e x)^{15/2}}{15 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^8 + (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*
(d + e*x)^(3/2))/(3*e^8) - (6*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(5
/2))/(5*e^8) - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(7
/2))/(7*e^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(9/2))/(9*e^8) + (6*c^2
*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(11/2))/(11*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(13/2))/(13*e^8
) + (2*B*c^3*(d + e*x)^(15/2))/(15*e^8)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 \sqrt{d+e x}}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) \sqrt{d+e x}}{e^7}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{3/2}}{e^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^{5/2}}{e^7}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^{7/2}}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{9/2}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{11/2}}{e^7}+\frac{B c^3 (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (B d-A e) \left (c d^2+a e^2\right )^3 \sqrt{d+e x}}{e^8}+\frac{2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^8}-\frac{6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{5/2}}{5 e^8}-\frac{2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{7/2}}{7 e^8}-\frac{2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{9/2}}{9 e^8}+\frac{6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{11/2}}{11 e^8}-\frac{2 c^3 (7 B d-A e) (d+e x)^{13/2}}{13 e^8}+\frac{2 B c^3 (d+e x)^{15/2}}{15 e^8}\\ \end{align*}

Mathematica [A]  time = 0.38074, size = 373, normalized size = 1.07 \[ \frac{2 \sqrt{d+e x} \left (3 A e \left (3003 a^2 c e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+15015 a^3 e^6+143 a c^2 e^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-512 d^5 e x+1024 d^6-252 d e^5 x^5+231 e^6 x^6\right )\right )+B \left (3861 a^2 c e^4 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+15015 a^3 e^6 (e x-2 d)+195 a c^2 e^2 \left (-96 d^3 e^2 x^2+80 d^2 e^3 x^3+128 d^4 e x-256 d^5-70 d e^4 x^4+63 e^5 x^5\right )-7 c^3 \left (768 d^5 e^2 x^2-640 d^4 e^3 x^3+560 d^3 e^4 x^4-504 d^2 e^5 x^5-1024 d^6 e x+2048 d^7+462 d e^6 x^6-429 e^7 x^7\right )\right )\right )}{45045 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(3*A*e*(15015*a^3*e^6 + 3003*a^2*c*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 143*a*c^2*e^2*(128*d^4
- 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2
- 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)) + B*(15015*a^3*e^6*(-2*d + e*x) + 3861*a^2
*c*e^4*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 195*a*c^2*e^2*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^
2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) - 7*c^3*(2048*d^7 - 1024*d^6*e*x + 768*d^5*e^2*x^2 - 640*d^4*e
^3*x^3 + 560*d^3*e^4*x^4 - 504*d^2*e^5*x^5 + 462*d*e^6*x^6 - 429*e^7*x^7))))/(45045*e^8)

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Maple [A]  time = 0.007, size = 489, normalized size = 1.4 \begin{align*}{\frac{6006\,B{c}^{3}{x}^{7}{e}^{7}+6930\,A{c}^{3}{e}^{7}{x}^{6}-6468\,B{c}^{3}d{e}^{6}{x}^{6}-7560\,A{c}^{3}d{e}^{6}{x}^{5}+24570\,Ba{c}^{2}{e}^{7}{x}^{5}+7056\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}+30030\,Aa{c}^{2}{e}^{7}{x}^{4}+8400\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}-27300\,Ba{c}^{2}d{e}^{6}{x}^{4}-7840\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}-34320\,Aa{c}^{2}d{e}^{6}{x}^{3}-9600\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}+38610\,B{a}^{2}c{e}^{7}{x}^{3}+31200\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}+8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}+54054\,A{a}^{2}c{e}^{7}{x}^{2}+41184\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}+11520\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}-46332\,B{a}^{2}cd{e}^{6}{x}^{2}-37440\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}-10752\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}-72072\,A{a}^{2}cd{e}^{6}x-54912\,Aa{c}^{2}{d}^{3}{e}^{4}x-15360\,A{c}^{3}{d}^{5}{e}^{2}x+30030\,B{a}^{3}{e}^{7}x+61776\,B{a}^{2}c{d}^{2}{e}^{5}x+49920\,Ba{c}^{2}{d}^{4}{e}^{3}x+14336\,B{c}^{3}{d}^{6}ex+90090\,A{a}^{3}{e}^{7}+144144\,A{a}^{2}c{d}^{2}{e}^{5}+109824\,Aa{c}^{2}{d}^{4}{e}^{3}+30720\,A{c}^{3}{d}^{6}e-60060\,B{a}^{3}d{e}^{6}-123552\,B{a}^{2}c{d}^{3}{e}^{4}-99840\,Ba{c}^{2}{d}^{5}{e}^{2}-28672\,B{c}^{3}{d}^{7}}{45045\,{e}^{8}}\sqrt{ex+d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(1/2)*(3003*B*c^3*e^7*x^7+3465*A*c^3*e^7*x^6-3234*B*c^3*d*e^6*x^6-3780*A*c^3*d*e^6*x^5+12285*B
*a*c^2*e^7*x^5+3528*B*c^3*d^2*e^5*x^5+15015*A*a*c^2*e^7*x^4+4200*A*c^3*d^2*e^5*x^4-13650*B*a*c^2*d*e^6*x^4-392
0*B*c^3*d^3*e^4*x^4-17160*A*a*c^2*d*e^6*x^3-4800*A*c^3*d^3*e^4*x^3+19305*B*a^2*c*e^7*x^3+15600*B*a*c^2*d^2*e^5
*x^3+4480*B*c^3*d^4*e^3*x^3+27027*A*a^2*c*e^7*x^2+20592*A*a*c^2*d^2*e^5*x^2+5760*A*c^3*d^4*e^3*x^2-23166*B*a^2
*c*d*e^6*x^2-18720*B*a*c^2*d^3*e^4*x^2-5376*B*c^3*d^5*e^2*x^2-36036*A*a^2*c*d*e^6*x-27456*A*a*c^2*d^3*e^4*x-76
80*A*c^3*d^5*e^2*x+15015*B*a^3*e^7*x+30888*B*a^2*c*d^2*e^5*x+24960*B*a*c^2*d^4*e^3*x+7168*B*c^3*d^6*e*x+45045*
A*a^3*e^7+72072*A*a^2*c*d^2*e^5+54912*A*a*c^2*d^4*e^3+15360*A*c^3*d^6*e-30030*B*a^3*d*e^6-61776*B*a^2*c*d^3*e^
4-49920*B*a*c^2*d^5*e^2-14336*B*c^3*d^7)/e^8

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Maxima [A]  time = 1.01297, size = 612, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} B c^{3} - 3465 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 12285 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 27027 \,{\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 45045 \,{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \sqrt{e x + d}\right )}}{45045 \, e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*c^3 - 3465*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(13/2) + 12285*(7*B*c^3*d^2 - 2*A*
c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(11/2) - 5005*(35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e
^3)*(e*x + d)^(9/2) + 6435*(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*
e^4)*(e*x + d)^(7/2) - 27027*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c
*d*e^4 - A*a^2*c*e^5)*(e*x + d)^(5/2) + 15015*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d
^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5 + B*a^3*e^6)*(e*x + d)^(3/2) - 45045*(B*c^3*d^7 - A*c^3*d^6*e + 3
*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)*sqrt(e
*x + d))/e^8

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Fricas [A]  time = 1.54625, size = 1100, normalized size = 3.16 \begin{align*} \frac{2 \,{\left (3003 \, B c^{3} e^{7} x^{7} - 14336 \, B c^{3} d^{7} + 15360 \, A c^{3} d^{6} e - 49920 \, B a c^{2} d^{5} e^{2} + 54912 \, A a c^{2} d^{4} e^{3} - 61776 \, B a^{2} c d^{3} e^{4} + 72072 \, A a^{2} c d^{2} e^{5} - 30030 \, B a^{3} d e^{6} + 45045 \, A a^{3} e^{7} - 231 \,{\left (14 \, B c^{3} d e^{6} - 15 \, A c^{3} e^{7}\right )} x^{6} + 63 \,{\left (56 \, B c^{3} d^{2} e^{5} - 60 \, A c^{3} d e^{6} + 195 \, B a c^{2} e^{7}\right )} x^{5} - 35 \,{\left (112 \, B c^{3} d^{3} e^{4} - 120 \, A c^{3} d^{2} e^{5} + 390 \, B a c^{2} d e^{6} - 429 \, A a c^{2} e^{7}\right )} x^{4} + 5 \,{\left (896 \, B c^{3} d^{4} e^{3} - 960 \, A c^{3} d^{3} e^{4} + 3120 \, B a c^{2} d^{2} e^{5} - 3432 \, A a c^{2} d e^{6} + 3861 \, B a^{2} c e^{7}\right )} x^{3} - 3 \,{\left (1792 \, B c^{3} d^{5} e^{2} - 1920 \, A c^{3} d^{4} e^{3} + 6240 \, B a c^{2} d^{3} e^{4} - 6864 \, A a c^{2} d^{2} e^{5} + 7722 \, B a^{2} c d e^{6} - 9009 \, A a^{2} c e^{7}\right )} x^{2} +{\left (7168 \, B c^{3} d^{6} e - 7680 \, A c^{3} d^{5} e^{2} + 24960 \, B a c^{2} d^{4} e^{3} - 27456 \, A a c^{2} d^{3} e^{4} + 30888 \, B a^{2} c d^{2} e^{5} - 36036 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^3*e^7*x^7 - 14336*B*c^3*d^7 + 15360*A*c^3*d^6*e - 49920*B*a*c^2*d^5*e^2 + 54912*A*a*c^2*d^4*
e^3 - 61776*B*a^2*c*d^3*e^4 + 72072*A*a^2*c*d^2*e^5 - 30030*B*a^3*d*e^6 + 45045*A*a^3*e^7 - 231*(14*B*c^3*d*e^
6 - 15*A*c^3*e^7)*x^6 + 63*(56*B*c^3*d^2*e^5 - 60*A*c^3*d*e^6 + 195*B*a*c^2*e^7)*x^5 - 35*(112*B*c^3*d^3*e^4 -
120*A*c^3*d^2*e^5 + 390*B*a*c^2*d*e^6 - 429*A*a*c^2*e^7)*x^4 + 5*(896*B*c^3*d^4*e^3 - 960*A*c^3*d^3*e^4 + 312
0*B*a*c^2*d^2*e^5 - 3432*A*a*c^2*d*e^6 + 3861*B*a^2*c*e^7)*x^3 - 3*(1792*B*c^3*d^5*e^2 - 1920*A*c^3*d^4*e^3 +
6240*B*a*c^2*d^3*e^4 - 6864*A*a*c^2*d^2*e^5 + 7722*B*a^2*c*d*e^6 - 9009*A*a^2*c*e^7)*x^2 + (7168*B*c^3*d^6*e -
7680*A*c^3*d^5*e^2 + 24960*B*a*c^2*d^4*e^3 - 27456*A*a*c^2*d^3*e^4 + 30888*B*a^2*c*d^2*e^5 - 36036*A*a^2*c*d*
e^6 + 15015*B*a^3*e^7)*x)*sqrt(e*x + d)/e^8

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Sympy [A]  time = 105.378, size = 1284, normalized size = 3.69 \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)*(c*x**2+a)**3/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*a**3*d/sqrt(d + e*x) + 2*A*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 6*A*a**2*c*d*(d**2/sqrt(
d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*A*a**2*c*(-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 + 6*A*a*c**2*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 + 6*A*a*c**2*(-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*A*c**3*d*(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) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 + 2*A*c*
*3*(-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d
+ e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6 + 2*B*a**3*
d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*B*a**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3
)/e + 6*B*a**2*c*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
+ 6*B*a**2*c*(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 + 6*B*a*c**2*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 + 6*B*a*c**2*(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) - 15*d**2*(d + e*x)**(7/2)/7 +
2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5 + 2*B*c**3*d*(-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e*x) +
7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d
*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**7 + 2*B*c**3*(d**8/sqrt(d + e*x) + 8*d**7*sqrt(d + e*x) - 28*
d**6*(d + e*x)**(3/2)/3 + 56*d**5*(d + e*x)**(5/2)/5 - 10*d**4*(d + e*x)**(7/2) + 56*d**3*(d + e*x)**(9/2)/9 -
28*d**2*(d + e*x)**(11/2)/11 + 8*d*(d + e*x)**(13/2)/13 - (d + e*x)**(15/2)/15)/e**7)/e, Ne(e, 0)), ((A*a**3*
x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x**2/2 + 3*B*a**2*c*x**4/4 + B*a*c**2*x**6/2 +
B*c**3*x**8/8)/sqrt(d), True))

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Giac [A]  time = 1.26351, size = 680, normalized size = 1.95 \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)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/
2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*c*e^(-2) + 3861*(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)*B*a^2*c*e^(-3) + 429*(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*a*c^2*e^(-4) + 195*(63*(x*e + d)^(11/2) - 38
5*(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*a*c^2*e^(-5) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2
- 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*c
^3*e^(-6) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(
9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e +
d)*d^7)*B*c^3*e^(-7) + 45045*sqrt(x*e + d)*A*a^3)*e^(-1)